3. Multiplying and Dividing Algebraic Fractions

Multiplying and Dividing Algebraic Fractions

Algebraic fractions can be multiplied and divided using the same basic rules as numerical fractions.

For multiplication:

a/b × c/d = ac/bd

For division, multiply by the reciprocal of the second fraction:

a/b ÷ c/d = a/b × d/c

The most efficient method is usually:

Restrict → Factor → Convert division → Cancel → Multiply → Simplify

Factoring before multiplying often prevents large, complicated expressions.

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Multiplying Numerical Fractions

Start with an ordinary fraction:

2/3 × 5/7

Multiply the numerators:

2 × 5 = 10

Multiply the denominators:

3 × 7 = 21

Therefore:

2/3 × 5/7 = 10/21

The same rule applies when variables are involved.

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Multiplying Algebraic Fractions

Consider:

x/3 × 5/7

Multiply the numerators:

x × 5 = 5x

Multiply the denominators:

3 × 7 = 21

Therefore:

5x/21

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Variables in Both Fractions

Multiply:

2x/5 × 3y/4

Numerator:

(2x)(3y) = 6xy

Denominator:

(5)(4) = 20

So:

6xy/20

Simplify:

3xy/10

Therefore:

2x/5 × 3y/4 = 3xy/10

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Simplify Before Multiplying

Instead of multiplying first and simplifying afterward, it is often easier to cancel common factors first.

Consider:

4x/9 × 3/8x

Restrictions:

x ≠ 0

Factor the numbers:

4x/9 × 3/(8x)

Cancel x.

Simplify:

4/8 = 1/2

and:

3/9 = 1/3

Now multiply:

1/3 × 1/2 = 1/6

Therefore:

4x/9 × 3/(8x) = 1/6, where x ≠ 0

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Why Cancelling First Helps

Suppose we multiply:

12x/35 × 14/9x

We could multiply first:

168x/(315x)

and then simplify.

But it is easier to simplify first.

Cancel x.

Simplify:

12/9 = 4/3

and:

14/35 = 2/5

Now:

4/5 × 2/3

Therefore:

8/15

Factoring and cancelling first keeps the numbers manageable.

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Cancellation Works Across Multiplication

When fractions are multiplied, the entire expression is a product.

For example:

2x/3 × 9/(4x)

can be viewed as:

(2 × x × 9)/(3 × 4 × x)

Now common factors can be cancelled.

Cancel x:

(2 × 9)/(3 × 4)

Simplify:

2/4 = 1/2

9/3 = 3

Therefore:

3/2

with:

x ≠ 0

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Cancel Factors, Not Terms

The same rule from simplifying algebraic fractions still applies.

You may cancel factors.

You may not cancel individual terms separated by addition or subtraction.

For example:

(x + 3)/x × x/5

Here x can cancel between the denominator of the first fraction and numerator of the second:

(x + 3)/5

But the x inside:

x + 3

cannot be cancelled.

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Factor Before Multiplying

Consider:

(x² − 9)/(x + 2) × (x + 2)/(x − 3)

First factor:

x² − 9 = (x − 3)(x + 3)

So:

[(x − 3)(x + 3)]/(x + 2) × (x + 2)/(x − 3)

Cancel:

x + 2

and:

x − 3

Result:

x + 3

Restrictions from the original expression:

x ≠ −2

x ≠ 3

Therefore:

x + 3, where x ≠ −2, 3

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Restrictions Come from the Original Expression

Consider:

(x + 4)/(x − 2) × (x − 2)/(x + 1)

Cancel x − 2:

(x + 4)/(x + 1)

But the original denominators were:

x − 2

and:

x + 1

Therefore:

x ≠ 2

and:

x ≠ −1

The restriction x ≠ 2 remains even though x − 2 cancelled.

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Multiplying Binomial Fractions

Multiply:

(x + 2)/(x + 5) × (x + 5)/(x + 7)

Restrictions:

x ≠ −5, −7

Cancel x + 5:

(x + 2)/(x + 7)

Therefore:

(x + 2)/(x + 7), where x ≠ −5, −7

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Factoring Common Factors First

Multiply:

(2x + 6)/(x + 4) × (x + 4)/(4x)

Restrictions:

x ≠ −4, 0

Factor:

2(x + 3)/(x + 4) × (x + 4)/(4x)

Cancel x + 4.

Simplify 2/4:

1/2

Result:

(x + 3)/(2x)

Therefore:

(x + 3)/(2x), where x ≠ −4, 0

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Factoring Trinomials Before Multiplying

Multiply:

(x² + 5x + 6)/(x + 4) × (x + 4)/(x + 2)

Factor:

x² + 5x + 6 = (x + 2)(x + 3)

So:

[(x + 2)(x + 3)]/(x + 4) × (x + 4)/(x + 2)

Cancel:

x + 4

and:

x + 2

Result:

x + 3

Restrictions:

x ≠ −4, −2

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Multiplying Expressions with Powers

Multiply:

x²/6 × 9/x

Restriction:

x ≠ 0

Simplify:

x²/x = x

and:

9/6 = 3/2

Therefore:

3x/2

where:

x ≠ 0

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Multiple Variables

Multiply:

3x²/4y × 8y²/9x

Restrictions:

x ≠ 0

y ≠ 0

Simplify:

3/9 = 1/3

8/4 = 2

x²/x = x

y²/y = y

Therefore:

2xy/3

with:

x ≠ 0, y ≠ 0

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Dividing Fractions

To divide fractions, multiply by the reciprocal of the second fraction.

For numerical fractions:

2/3 ÷ 5/7

Flip the second fraction:

2/3 × 7/5

Multiply:

14/15

Therefore:

2/3 ÷ 5/7 = 14/15

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What Is a Reciprocal?

The reciprocal of a fraction is found by exchanging its numerator and denominator.

The reciprocal of:

3/5

is:

5/3

The reciprocal of:

x/7

is:

7/x

The reciprocal of:

(x + 2)/(x − 4)

is:

(x − 4)/(x + 2)

provided the required restrictions are satisfied.

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Why Reciprocals Work

A number multiplied by its reciprocal equals 1.

For example:

3/5 × 5/3 = 1

Similarly:

x/4 × 4/x = 1

provided:

x ≠ 0

This is why division by a fraction can be rewritten as multiplication by its reciprocal.

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Dividing Algebraic Fractions

Consider:

x/3 ÷ 2/5

Keep the first fraction:

x/3

Change division to multiplication:

×

Take the reciprocal of the second fraction:

5/2

So:

x/3 × 5/2

Multiply:

5x/6

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A Useful Memory Aid

For division:

Keep → Change → Flip

Keep the first fraction.

Change ÷ to ×.

Flip the second fraction.

However, understanding the reciprocal rule is more important than simply memorizing the phrase.

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Dividing with Variables

Simplify:

3x/4 ÷ 6x/5

Restrictions initially include:

x ≠ 0

Rewrite:

3x/4 × 5/(6x)

Cancel x.

Simplify 3/6:

1/2

Therefore:

5/8

with:

x ≠ 0

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Dividing Binomial Fractions

Simplify:

(x + 2)/(x − 3) ÷ (x + 2)/(x + 5)

First rewrite using the reciprocal:

(x + 2)/(x − 3) × (x + 5)/(x + 2)

Cancel x + 2:

(x + 5)/(x − 3)

Now consider restrictions carefully.

From the original denominators:

x ≠ 3

x ≠ −5

But because we are dividing by:

(x + 2)/(x + 5)

the divisor itself cannot equal zero.

Therefore:

x + 2 ≠ 0

so:

x ≠ −2

Final result:

(x + 5)/(x − 3), where x ≠ 3, −5, −2

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Division Has an Extra Restriction

This is important.

Consider:

A ÷ B

Not only must denominators be nonzero, but:

B ≠ 0

because division by zero is undefined.

For example:

5/x ÷ (x − 3)/(x + 2)

Restrictions from denominators:

x ≠ 0

x ≠ −2

The divisor must also not equal zero:

x − 3 ≠ 0

Therefore:

x ≠ 3

So the complete restrictions are:

x ≠ 0, −2, 3

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A Complete Division Example

Simplify:

(x² − 9)/(x + 4) ÷ (x − 3)/(x + 2)

Restrictions from original denominators:

x ≠ −4

x ≠ −2

The divisor cannot equal zero:

x − 3 ≠ 0

so:

x ≠ 3

Now convert division to multiplication:

(x² − 9)/(x + 4) × (x + 2)/(x − 3)

Factor:

x² − 9 = (x − 3)(x + 3)

So:

(x − 3)(x + 3)/(x + 4) × (x + 2)/(x − 3)

Cancel x − 3:

(x + 3)(x + 2)/(x + 4)

Final answer:

(x + 3)(x + 2)/(x + 4)

where:

x ≠ −4, −2, 3

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Never Flip the First Fraction

For:

a/b ÷ c/d

the correct conversion is:

a/b × d/c

Only the second fraction is replaced by its reciprocal.

Incorrect:

b/a × d/c

Do not flip both fractions.

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Never Flip During Multiplication

If the operation is already multiplication:

a/b × c/d

do not take any reciprocal.

Simply factor, cancel common factors, and multiply.

Reciprocals are specifically used to convert division into multiplication.

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Factoring Before Division

Simplify:

(x² − 4)/(x + 3) ÷ (x − 2)/(x + 5)

Restrictions:

x ≠ −3

x ≠ −5

The divisor cannot equal zero:

x ≠ 2

Convert:

(x² − 4)/(x + 3) × (x + 5)/(x − 2)

Factor:

x² − 4 = (x − 2)(x + 2)

So:

(x − 2)(x + 2)/(x + 3) × (x + 5)/(x − 2)

Cancel x − 2:

(x + 2)(x + 5)/(x + 3)

Therefore:

(x + 2)(x + 5)/(x + 3), where x ≠ −3, −5, 2

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Factoring Both Fractions

Simplify:

(x² + 5x + 6)/(x² − 9) × (x − 3)/(x + 2)

Factor:

x² + 5x + 6 = (x + 2)(x + 3)

x² − 9 = (x − 3)(x + 3)

So:

(x + 2)(x + 3)/[(x − 3)(x + 3)] × (x − 3)/(x + 2)

Cancel:

x + 2

x + 3

x − 3

Result:

1

Restrictions from the original denominators:

x ≠ 3, −3, −2

Therefore:

1, where x ≠ −3, −2, 3

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Multi-Step Fraction Problems

Some expressions contain several multiplication and division operations.

For example:

x/3 × 6/(x + 2) ÷ 2x/(x + 2)

Work systematically.

First convert division to multiplication:

x/3 × 6/(x + 2) × (x + 2)/(2x)

Now cancel common factors:

• x cancels
• x + 2 cancels
• 6/(3 × 2) = 1

Therefore:

1

Restrictions include:

x ≠ 0

x ≠ −2

The divisor:

2x/(x + 2)

must also not equal zero, which again requires:

x ≠ 0

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Multiple Division Operations

Consider:

a/b ÷ c/d ÷ e/f

Division is performed from left to right.

First:

a/b ÷ c/d = a/b × d/c

Then divide the result by e/f:

a/b × d/c × f/e

So:

adf/bce

assuming all required quantities are nonzero.

Do not simply flip every fraction without considering the operations.

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Fractions Containing Constants

Simplify:

5x/12 × 18/(10x)

Restriction:

x ≠ 0

Simplify:

5/10 = 1/2

18/12 = 3/2

Cancel x.

Therefore:

3/4

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Fractions with Negative Signs

Consider:

−2x/5 × 15/(4x)

Restriction:

x ≠ 0

Cancel x.

Simplify:

15/5 = 3

and:

−2/4 = −1/2

Therefore:

−3/2

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Negative Signs Can Be Written in Different Places

These are equivalent:

−a/b

(−a)/b

a/(−b)

But:

(−a)/(−b) = a/b

Two negative factors produce a positive result.

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Reversed Factors

Consider:

(x − 4)/(x + 2) × (x + 2)/(4 − x)

Notice:

4 − x = −(x − 4)

So:

(x − 4)/(x + 2) × (x + 2)/[−(x − 4)]

Cancel:

x + 2

and:

x − 4

Result:

−1

Restrictions:

x ≠ −2, 4

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Complex Fractions

A complex fraction is a fraction containing another fraction.

For example:

(x/3)/(2/5)

This means:

x/3 ÷ 2/5

Convert:

x/3 × 5/2

Therefore:

5x/6

Complex fractions can often be handled by rewriting the large fraction bar as division.

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Another Complex Fraction

Consider:

[(x + 2)/4]/[(x − 1)/6]

Rewrite:

(x + 2)/4 ÷ (x − 1)/6

Multiply by the reciprocal:

(x + 2)/4 × 6/(x − 1)

Simplify:

6/4 = 3/2

Therefore:

3(x + 2)/[2(x − 1)]

Restrictions:

x ≠ 1

The divisor must also be nonzero:

x +?

Here the divisor is:

(x − 1)/6

which is zero when x = 1, already excluded.

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A Multi-Step Factoring Example

Simplify:

(x² − 16)/(x² + 7x + 12) × (x + 3)/(x − 4)

Factor:

x² − 16 = (x − 4)(x + 4)

x² + 7x + 12 = (x + 3)(x + 4)

So:

(x − 4)(x + 4)/[(x + 3)(x + 4)] × (x + 3)/(x − 4)

Cancel all common factors:

1

Restrictions:

From:

(x + 3)(x + 4)

we get:

x ≠ −3, −4

From:

x − 4

we get:

x ≠ 4

Therefore:

1, where x ≠ −4, −3, 4

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A Multi-Step Division Example

Simplify:

(x² − 25)/(x² + 3x − 10) ÷ (x + 5)/(x − 2)

Factor the first denominator:

x² + 3x − 10 = (x + 5)(x − 2)

Restrictions from denominators:

x ≠ −5, 2

The divisor:

(x + 5)/(x − 2)

cannot equal zero, so:

x ≠ −5

Now convert division:

(x² − 25)/[(x + 5)(x − 2)] × (x − 2)/(x + 5)

Factor:

x² − 25 = (x − 5)(x + 5)

Then:

(x − 5)(x + 5)/[(x + 5)(x − 2)] × (x − 2)/(x + 5)

Cancel x − 2.

Cancel one x + 5.

Result:

(x − 5)/(x + 5)

Therefore:

(x − 5)/(x + 5), where x ≠ −5, 2

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Verifying by Substitution

Suppose:

(x² − 9)/(x + 2) × (x + 2)/(x − 3)

simplifies to:

x + 3

with:

x ≠ −2, 3

Choose an allowed value:

x = 5

Original:

(25 − 9)/7 × 7/2

= 16/7 × 7/2

= 8

Simplified:

5 + 3 = 8

The results agree.

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Verification Can Catch Cancellation Errors

Suppose someone claims:

(x + 4)/x × 2/3 = 8/3

by cancelling x incorrectly.

Choose:

x = 2

Original:

(2 + 4)/2 × 2/3

= 6/2 × 2/3

= 3 × 2/3

= 2

Claimed answer:

8/3

Since:

2 ≠ 8/3

the simplification is incorrect.

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When an Answer Cannot Be Simplified Further

Consider:

(x + 3)/(x + 5) × (x + 7)/(x + 2)

There are no common factors between numerators and denominators.

Therefore:

(x + 3)(x + 7)/[(x + 5)(x + 2)]

is already simplified.

It is usually best to leave it in factored form.

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Factored Form Is Often Best

When working with algebraic fractions, factored form makes restrictions and common factors easier to see.

For example:

(x − 3)(x + 2)/[(x + 5)(x − 7)]

is often more useful than expanding everything.

Factored form helps us see immediately that the restrictions are:

x ≠ −5, 7

and whether any common factors can be cancelled.

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Algebraic Fractions in Science

Multiplication and division of algebraic fractions appear in scientific formulas.

For example:

v = d/t

and:

t = d/v

Suppose another relationship gives:

d = Pt/F

Substituting this into the speed equation gives:

v = (Pt/F)/t

This can be viewed as:

v = Pt/(Ft)

Cancel t:

v = P/F

where the relevant denominators must be nonzero.

Algebraic fraction skills therefore become increasingly useful when formulas are combined.

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Solving Practical Problems

Suppose fuel efficiency is modeled by:

E = d/f

where:

• E = distance traveled per unit of fuel
• d = distance
• f = fuel used

If another relationship gives:

d = vt

then substitute:

E = vt/f

If:

v = 80 km/h

t = 4 h

f = 25 L

then:

E = (80)(4)/25

E = 320/25

Therefore:

E = 12.8 km/L

This combines substitution, multiplication, and division.

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Common Mistakes

Mistake: Adding numerators or denominators when multiplying.

Incorrect:

a/b × c/d = (a + c)/(b + d)

Correct:

a/b × c/d = ac/bd

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Mistake: Flipping the second fraction during multiplication.

Reciprocals are used for division, not ordinary multiplication.

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Mistake: Flipping the first fraction during division.

For:

a/b ÷ c/d

only the second fraction is replaced by its reciprocal.

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Mistake: Forgetting to change ÷ to ×.

The reciprocal rule requires:

÷ → × reciprocal

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Mistake: Cancelling terms instead of factors.

You cannot cancel x from:

(x + 3)/x

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Mistake: Multiplying before factoring.

This often creates unnecessarily complicated expressions.

Factor first whenever possible.

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Mistake: Forgetting restrictions.

Cancelled factors still produce restrictions if they appeared in an original denominator.

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Mistake: Forgetting that a divisor cannot equal zero.

When dividing by an algebraic fraction, the entire divisor must be nonzero.

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Mistake: Expanding the final answer unnecessarily.

Factored form is often simpler and clearer.

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Mistake: Cancelling factors before division has been converted to multiplication.

First replace division with multiplication by the reciprocal. Then cancel.

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Key Terms

Algebraic fraction — fraction containing variables or algebraic expressions.

Numerator — expression above the fraction line.

Denominator — expression below the fraction line.

Factor — quantity multiplied by another quantity.

Common factor — factor shared between numerator and denominator.

Factorization — rewriting an expression as a product of factors.

Cancel — divide numerator and denominator by the same nonzero factor.

Reciprocal — fraction formed by exchanging numerator and denominator.

Restriction — value excluded because it makes an original denominator zero or otherwise makes an operation undefined.

Product — result of multiplication.

Quotient — result of division.

Complex fraction — fraction containing one or more fractions within its numerator or denominator.

Difference of squares — factoring pattern a² − b² = (a − b)(a + b).

Equivalent expression — expression with the same value over its permitted domain.

Verification — process of checking that a result is mathematically equivalent to the original.

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Key Takeaways

• Algebraic fractions multiply using the same basic rule as numerical fractions.
• Multiply numerator by numerator and denominator by denominator.
• Factor expressions before multiplying whenever possible.
• Cancelling before multiplication often makes calculations much easier.
• Only common factors can be cancelled.
• Terms separated by addition or subtraction cannot simply be cancelled.
• When dividing fractions, multiply by the reciprocal of the second fraction.
• A useful memory aid is Keep → Change → Flip.
• Only the divisor is replaced by its reciprocal.
• Do not use reciprocals when the operation is already multiplication.
• Factor after converting division to multiplication so common factors are visible.
• Difference-of-squares and trinomial factoring can reveal common factors.
• Restrictions from original denominators must be retained.
• Division introduces an additional requirement: the entire divisor cannot equal zero.
• Cancelled factors can still create restrictions.
• Factored form is often the clearest final form.
• Multi-step problems should be handled systematically rather than trying to perform every operation at once.
• Complex fractions can be interpreted as division problems.
• Negative signs must be handled carefully.
• Reversed factors such as x − a and a − x differ by a factor of −1.
• Substitution can be used to verify simplified answers.
• Multiplying and dividing algebraic fractions is important for later work with rational equations, functions, formulas, rates, and scientific relationships.

Check Your Understanding

1. Multiply and simplify 2x/3 × 5/4.
2. Multiply and simplify 3x/5 × 10/9x and state the restriction.
3. Multiply and simplify 4x²/7 × 14/8x.
4. Multiply and simplify 3x²/4y × 8y²/9x and state all restrictions.
5. Simplify (x + 2)/(x + 5) × (x + 5)/(x + 7).
6. Simplify (x² − 9)/(x + 4) × (x + 4)/(x − 3) and state all restrictions.
7. Simplify (x² + 5x + 6)/(x + 4) × (x + 4)/(x + 2).
8. Explain why factoring before multiplying algebraic fractions is often useful.
9. Find the reciprocal of 3x/5.
10. Find the reciprocal of (x + 2)/(x − 4).
11. Simplify x/4 ÷ 3/5.
12. Simplify 3x/7 ÷ 6x/5 and state the restriction.
13. Simplify (x + 3)/(x − 2) ÷ (x + 3)/(x + 5) and state all restrictions.
14. Simplify (x² − 16)/(x + 1) ÷ (x − 4)/(x + 3) and state all restrictions.
15. Explain why the fraction you are dividing by cannot equal zero.
16. Explain why x cannot be cancelled from (x + 5)/x.
17. Simplify (x − 6)/(x + 2) × (x + 2)/(6 − x).
18. Simplify [(x + 4)/5]/[(x − 2)/10] and state all restrictions.
19. Use substitution to explain how you could verify a simplified algebraic fraction.
20. Simplify completely:

(x² − 25)/(x² + 2x − 15) ÷ (x + 5)/(x − 3)

State all restrictions and explain each step.