Algebra Examples

${(5 x^{4} y^{3})}^{- 2} {(2 x^{3} y^{5} z^{8})}^{4} (y^{7} z^{- 3})$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(5 x^{4} y^{3})}^{2}} {(2 x^{3} y^{5} z^{8})}^{4} (y^{7} z^{- 3})$

Step 2

Simplify the denominator.

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Step 2.1

Apply the product rule to $5 x^{4} y^{3}$ .

$\frac{1}{{(5 x^{4})}^{2} {(y^{3})}^{2}} {(2 x^{3} y^{5} z^{8})}^{4} (y^{7} z^{- 3})$

Step 2.2

Apply the product rule to $5 x^{4}$ .

$\frac{1}{5^{2} {(x^{4})}^{2} {(y^{3})}^{2}} {(2 x^{3} y^{5} z^{8})}^{4} (y^{7} z^{- 3})$

Step 2.3

Raise $5$ to the power of $2$ .

$\frac{1}{25 {(x^{4})}^{2} {(y^{3})}^{2}} {(2 x^{3} y^{5} z^{8})}^{4} (y^{7} z^{- 3})$

Step 2.4

Multiply the exponents in ${(x^{4})}^{2}$ .

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Step 2.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{25 x^{4 \cdot 2} {(y^{3})}^{2}} {(2 x^{3} y^{5} z^{8})}^{4} (y^{7} z^{- 3})$

Step 2.4.2

Multiply $4$ by $2$ .

$\frac{1}{25 x^{8} {(y^{3})}^{2}} {(2 x^{3} y^{5} z^{8})}^{4} (y^{7} z^{- 3})$

Step 2.5

Multiply the exponents in ${(y^{3})}^{2}$ .

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Step 2.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{25 x^{8} y^{3 \cdot 2}} {(2 x^{3} y^{5} z^{8})}^{4} (y^{7} z^{- 3})$

Step 2.5.2

Multiply $3$ by $2$ .

$\frac{1}{25 x^{8} y^{6}} {(2 x^{3} y^{5} z^{8})}^{4} (y^{7} z^{- 3})$

Step 3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.1

Apply the product rule to $2 x^{3} y^{5} z^{8}$ .

$\frac{1}{25 x^{8} y^{6}} ({(2 x^{3} y^{5})}^{4} {(z^{8})}^{4}) (y^{7} z^{- 3})$

Step 3.2

Apply the product rule to $2 x^{3} y^{5}$ .

$\frac{1}{25 x^{8} y^{6}} ({(2 x^{3})}^{4} {(y^{5})}^{4} {(z^{8})}^{4}) (y^{7} z^{- 3})$

Step 3.3

Apply the product rule to $2 x^{3}$ .

$\frac{1}{25 x^{8} y^{6}} (2^{4} {(x^{3})}^{4} {(y^{5})}^{4} {(z^{8})}^{4}) (y^{7} z^{- 3})$

Step 4

Simplify terms.

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Step 4.1

Rewrite using the commutative property of multiplication.

$2^{4} \frac{1}{25 x^{8} y^{6}} ({(x^{3})}^{4} {(y^{5})}^{4} {(z^{8})}^{4}) (y^{7} z^{- 3})$

Step 4.2

Raise $2$ to the power of $4$ .

$16 \frac{1}{25 x^{8} y^{6}} ({(x^{3})}^{4} {(y^{5})}^{4} {(z^{8})}^{4}) (y^{7} z^{- 3})$

Step 4.3

Combine $16$ and $\frac{1}{25 x^{8} y^{6}}$ .

$\frac{16}{25 x^{8} y^{6}} ({(x^{3})}^{4} {(y^{5})}^{4} {(z^{8})}^{4}) (y^{7} z^{- 3})$

Step 4.4

Multiply the exponents in ${(x^{3})}^{4}$ .

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Step 4.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{16}{25 x^{8} y^{6}} (x^{3 \cdot 4} {(y^{5})}^{4} {(z^{8})}^{4}) (y^{7} z^{- 3})$

Step 4.4.2

Multiply $3$ by $4$ .

$\frac{16}{25 x^{8} y^{6}} (x^{12} {(y^{5})}^{4} {(z^{8})}^{4}) (y^{7} z^{- 3})$

Step 4.5

Cancel the common factor of $x^{8}$ .

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Step 4.5.1

Factor $x^{8}$ out of $25 x^{8} y^{6}$ .

$\frac{16}{x^{8} (25 y^{6})} (x^{12} {(y^{5})}^{4} {(z^{8})}^{4}) (y^{7} z^{- 3})$

Step 4.5.2

Factor $x^{8}$ out of $x^{12} {(y^{5})}^{4} {(z^{8})}^{4}$ .

$\frac{16}{x^{8} (25 y^{6})} (x^{8} (x^{4} {(y^{5})}^{4} {(z^{8})}^{4})) (y^{7} z^{- 3})$

Step 4.5.3

Cancel the common factor.

$\frac{16}{x^{8} (25 y^{6})} (x^{8} (x^{4} {(y^{5})}^{4} {(z^{8})}^{4})) (y^{7} z^{- 3})$

Step 4.5.4

Rewrite the expression.

$\frac{16}{25 y^{6}} (x^{4} {(y^{5})}^{4} {(z^{8})}^{4}) (y^{7} z^{- 3})$

Step 4.6

Combine $x^{4}$ and $\frac{16}{25 y^{6}}$ .

$\frac{x^{4} \cdot 16}{25 y^{6}} ({(y^{5})}^{4} {(z^{8})}^{4}) (y^{7} z^{- 3})$

Step 4.7

Combine ${(y^{5})}^{4}$ and $\frac{x^{4} \cdot 16}{25 y^{6}}$ .

$\frac{{(y^{5})}^{4} (x^{4} \cdot 16)}{25 y^{6}} {(z^{8})}^{4} (y^{7} z^{- 3})$

Step 4.8

Combine $\frac{{(y^{5})}^{4} (x^{4} \cdot 16)}{25 y^{6}}$ and ${(z^{8})}^{4}$ .

$\frac{{(y^{5})}^{4} (x^{4} \cdot 16) {(z^{8})}^{4}}{25 y^{6}} (y^{7} z^{- 3})$

Step 5

Simplify the numerator.

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Step 5.1

Multiply the exponents in ${(y^{5})}^{4}$ .

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Step 5.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{y^{5 \cdot 4} x^{4} \cdot 16 {(z^{8})}^{4}}{25 y^{6}} (y^{7} z^{- 3})$

Step 5.1.2

Multiply $5$ by $4$ .

$\frac{y^{20} x^{4} \cdot 16 {(z^{8})}^{4}}{25 y^{6}} (y^{7} z^{- 3})$

Step 5.2

Multiply the exponents in ${(z^{8})}^{4}$ .

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Step 5.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{y^{20} x^{4} \cdot 16 z^{8 \cdot 4}}{25 y^{6}} (y^{7} z^{- 3})$

Step 5.2.2

Multiply $8$ by $4$ .

$\frac{y^{20} x^{4} \cdot 16 z^{32}}{25 y^{6}} (y^{7} z^{- 3})$

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $y^{6}$ .

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Step 6.1.1

Factor $y^{6}$ out of $25 y^{6}$ .

$\frac{y^{20} x^{4} \cdot 16 z^{32}}{y^{6} \cdot 25} (y^{7} z^{- 3})$

Step 6.1.2

Factor $y^{6}$ out of $y^{7} z^{- 3}$ .

$\frac{y^{20} x^{4} \cdot 16 z^{32}}{y^{6} \cdot 25} (y^{6} (y z^{- 3}))$

Step 6.1.3

Cancel the common factor.

$\frac{y^{20} x^{4} \cdot 16 z^{32}}{y^{6} \cdot 25} (y^{6} (y z^{- 3}))$

Step 6.1.4

Rewrite the expression.

$\frac{y^{20} x^{4} \cdot 16 z^{32}}{25} (y z^{- 3})$

Step 6.2

Combine $y$ and $\frac{y^{20} x^{4} \cdot 16 z^{32}}{25}$ .

$\frac{y (y^{20} x^{4} \cdot 16 z^{32})}{25} z^{- 3}$

Step 7

Raise $y$ to the power of $1$ .

$\frac{y^{20} y^{1} (x^{4} \cdot 16 z^{32})}{25} z^{- 3}$

Step 8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{y^{20 + 1} (x^{4} \cdot 16 z^{32})}{25} z^{- 3}$

Step 9

Add $20$ and $1$ .

$\frac{y^{21} (x^{4} \cdot 16 z^{32})}{25} z^{- 3}$

Step 10

Combine $\frac{y^{21} (x^{4} \cdot 16 z^{32})}{25}$ and $z^{- 3}$ .

$\frac{y^{21} (x^{4} \cdot 16 z^{32}) z^{- 3}}{25}$

Step 11

Multiply $z^{32}$ by $z^{- 3}$ by adding the exponents.

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Step 11.1

Move $z^{- 3}$ .

$\frac{y^{21} (x^{4} \cdot 16 (z^{- 3} z^{32}))}{25}$

Step 11.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{y^{21} (x^{4} \cdot 16 z^{- 3 + 32})}{25}$

Step 11.3

Add $- 3$ and $32$ .

$\frac{y^{21} (x^{4} \cdot 16 z^{29})}{25}$

Step 12

Move $16$ to the left of $y^{21} x^{4}$ .

$\frac{16 y^{21} x^{4} z^{29}}{25}$