Algebra Examples

${(5^{\frac{5}{3}} \cdot 5^{\frac{7}{6}})}^{- 1}$

Step 1

Multiply $5^{\frac{5}{3}}$ by $5^{\frac{7}{6}}$ by adding the exponents.

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Step 1.1

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

${(5^{\frac{5}{3} + \frac{7}{6}})}^{- 1}$

Step 1.2

To write $\frac{5}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(5^{\frac{5}{3} \cdot \frac{2}{2} + \frac{7}{6}})}^{- 1}$

Step 1.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{5}{3}$ by $\frac{2}{2}$ .

${(5^{\frac{5 \cdot 2}{3 \cdot 2} + \frac{7}{6}})}^{- 1}$

Step 1.3.2

Multiply $3$ by $2$ .

${(5^{\frac{5 \cdot 2}{6} + \frac{7}{6}})}^{- 1}$

Step 1.4

Combine the numerators over the common denominator.

${(5^{\frac{5 \cdot 2 + 7}{6}})}^{- 1}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Multiply $5$ by $2$ .

${(5^{\frac{10 + 7}{6}})}^{- 1}$

Step 1.5.2

Add $10$ and $7$ .

${(5^{\frac{17}{6}})}^{- 1}$

Step 2

Multiply the exponents in ${(5^{\frac{17}{6}})}^{- 1}$ .

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Step 2.1

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

$5^{\frac{17}{6} \cdot - 1}$

Step 2.2

Multiply $\frac{17}{6} \cdot - 1$ .

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Step 2.2.1

Combine $\frac{17}{6}$ and $- 1$ .

$5^{\frac{17 \cdot - 1}{6}}$

Step 2.2.2

Multiply $17$ by $- 1$ .

$5^{\frac{- 17}{6}}$

Step 2.3

Move the negative in front of the fraction.

$5^{- \frac{17}{6}}$

Step 3

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

$\frac{1}{5^{\frac{17}{6}}}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{5^{\frac{17}{6}}}$

Decimal Form:

$0.01046128 \dots$