Basic Math Examples

${((a^{\frac{1}{2}} \cdot a^{\frac{1}{4}} \div a) \cdot a^{\frac{3}{2}})}^{4}$

Step 1

Multiply $a^{\frac{1}{2}}$ by $a^{\frac{1}{4}}$ 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.

${(a^{\frac{1}{2} + \frac{1}{4}} \div a \cdot a^{\frac{3}{2}})}^{4}$

Step 1.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(a^{\frac{1}{2} \cdot \frac{2}{2} + \frac{1}{4}} \div a \cdot a^{\frac{3}{2}})}^{4}$

Step 1.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

${(a^{\frac{2}{2 \cdot 2} + \frac{1}{4}} \div a \cdot a^{\frac{3}{2}})}^{4}$

Step 1.3.2

Multiply $2$ by $2$ .

${(a^{\frac{2}{4} + \frac{1}{4}} \div a \cdot a^{\frac{3}{2}})}^{4}$

Step 1.4

Combine the numerators over the common denominator.

${(a^{\frac{2 + 1}{4}} \div a \cdot a^{\frac{3}{2}})}^{4}$

Step 1.5

Add $2$ and $1$ .

${(a^{\frac{3}{4}} \div a \cdot a^{\frac{3}{2}})}^{4}$

Step 2

Rewrite the division as a fraction.

${(\frac{a^{\frac{3}{4}}}{a} \cdot a^{\frac{3}{2}})}^{4}$

Step 3

Move $a^{\frac{3}{4}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

${(\frac{1}{a \cdot a^{- \frac{3}{4}}} \cdot a^{\frac{3}{2}})}^{4}$

Step 4

Multiply $a$ by $a^{- \frac{3}{4}}$ by adding the exponents.

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Step 4.1

Multiply $a$ by $a^{- \frac{3}{4}}$ .

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Step 4.1.1

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

${(\frac{1}{a^{1} a^{- \frac{3}{4}}} \cdot a^{\frac{3}{2}})}^{4}$

Step 4.1.2

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

${(\frac{1}{a^{1 - \frac{3}{4}}} \cdot a^{\frac{3}{2}})}^{4}$

Step 4.2

Write $1$ as a fraction with a common denominator.

${(\frac{1}{a^{\frac{4}{4} - \frac{3}{4}}} \cdot a^{\frac{3}{2}})}^{4}$

Step 4.3

Combine the numerators over the common denominator.

${(\frac{1}{a^{\frac{4 - 3}{4}}} \cdot a^{\frac{3}{2}})}^{4}$

Step 4.4

Subtract $3$ from $4$ .

${(\frac{1}{a^{\frac{1}{4}}} \cdot a^{\frac{3}{2}})}^{4}$

Step 5

Cancel the common factor of $a^{\frac{1}{4}}$ .

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Step 5.1

Factor $a^{\frac{1}{4}}$ out of $a^{\frac{3}{2}}$ .

${(\frac{1}{a^{\frac{1}{4}}} \cdot (a^{\frac{1}{4}} a^{\frac{5}{4}}))}^{4}$

Step 5.2

Cancel the common factor.

${(\frac{1}{a^{\frac{1}{4}}} \cdot (a^{\frac{1}{4}} a^{\frac{5}{4}}))}^{4}$

Step 5.3

Rewrite the expression.

${(a^{\frac{5}{4}})}^{4}$

Step 6

Multiply the exponents in ${(a^{\frac{5}{4}})}^{4}$ .

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Step 6.1

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

$a^{\frac{5}{4} \cdot 4}$

Step 6.2

Cancel the common factor of $4$ .

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Step 6.2.1

Cancel the common factor.

$a^{\frac{5}{4} \cdot 4}$

Step 6.2.2

Rewrite the expression.

$a^{5}$