Basic Math Examples

{(\sqrt{\frac{3}{5}})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}

${(\sqrt{\frac{3}{5}})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1

Simplify each term.

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Step 1.1

Rewrite

$\sqrt{\frac{3}{5}}$ as

$\frac{\sqrt{3}}{\sqrt{5}}$ .

${(\frac{\sqrt{3}}{\sqrt{5}})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.2

Multiply

$\frac{\sqrt{3}}{\sqrt{5}}$ by

$\frac{\sqrt{5}}{\sqrt{5}}$ .

${(\frac{\sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply

$\frac{\sqrt{3}}{\sqrt{5}}$ by

$\frac{\sqrt{5}}{\sqrt{5}}$ .

${(\frac{\sqrt{3} \sqrt{5}}{\sqrt{5} \sqrt{5}})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.3.2

Raise

$\sqrt{5}$ to the power of

$1$ .

${(\frac{\sqrt{3} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.3.3

Raise

$\sqrt{5}$ to the power of

$1$ .

${(\frac{\sqrt{3} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.3.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{\sqrt{3} \sqrt{5}}{{\sqrt{5}}^{1 + 1}})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.3.5

Add

$1$ and

$1$ .

${(\frac{\sqrt{3} \sqrt{5}}{{\sqrt{5}}^{2}})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.3.6

Rewrite

${\sqrt{5}}^{2}$ as

$5$ .

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Step 1.3.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{5}$ as

$5^{\frac{1}{2}}$ .

${(\frac{\sqrt{3} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.3.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

${(\frac{\sqrt{3} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

${(\frac{\sqrt{3} \sqrt{5}}{5^{\frac{2}{2}}})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.3.6.4

Cancel the common factor of

$2$ .

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Step 1.3.6.4.1

Cancel the common factor.

${(\frac{\sqrt{3} \sqrt{5}}{5^{\frac{2}{2}}})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.3.6.4.2

Rewrite the expression.

${(\frac{\sqrt{3} \sqrt{5}}{5^{1}})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.3.6.5

Evaluate the exponent.

${(\frac{\sqrt{3} \sqrt{5}}{5})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Combine using the product rule for radicals.

${(\frac{\sqrt{3 \cdot 5}}{5})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.4.2

Multiply

$3$ by

$5$ .

${(\frac{\sqrt{15}}{5})}^{a} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.5

Apply the product rule to

$\frac{\sqrt{15}}{5}$ .

$\frac{{\sqrt{15}}^{a}}{5^{a}} - {(\sqrt{\frac{625}{81}})}^{a + 3}$

Step 1.6

Rewrite

$\sqrt{\frac{625}{81}}$ as

$\frac{\sqrt{625}}{\sqrt{81}}$ .

$\frac{{\sqrt{15}}^{a}}{5^{a}} - {(\frac{\sqrt{625}}{\sqrt{81}})}^{a + 3}$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Rewrite

$625$ as

$25^{2}$ .

$\frac{{\sqrt{15}}^{a}}{5^{a}} - {(\frac{\sqrt{25^{2}}}{\sqrt{81}})}^{a + 3}$

Step 1.7.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{{\sqrt{15}}^{a}}{5^{a}} - {(\frac{25}{\sqrt{81}})}^{a + 3}$

Step 1.8

Simplify the denominator.

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Step 1.8.1

Rewrite

$81$ as

$9^{2}$ .

$\frac{{\sqrt{15}}^{a}}{5^{a}} - {(\frac{25}{\sqrt{9^{2}}})}^{a + 3}$

Step 1.8.2

Pull terms out from under the radical, assuming positive real numbers.

$\frac{{\sqrt{15}}^{a}}{5^{a}} - {(\frac{25}{9})}^{a + 3}$

Step 1.9

Apply the product rule to

$\frac{25}{9}$ .

$\frac{{\sqrt{15}}^{a}}{5^{a}} - \frac{25^{a + 3}}{9^{a + 3}}$

Step 2

To write

$\frac{{\sqrt{15}}^{a}}{5^{a}}$ as a fraction with a common denominator, multiply by

$\frac{9^{a + 3}}{9^{a + 3}}$ .

$\frac{{\sqrt{15}}^{a}}{5^{a}} \cdot \frac{9^{a + 3}}{9^{a + 3}} - \frac{25^{a + 3}}{9^{a + 3}}$

Step 3

To write

$- \frac{25^{a + 3}}{9^{a + 3}}$ as a fraction with a common denominator, multiply by

$\frac{5^{a}}{5^{a}}$ .

$\frac{{\sqrt{15}}^{a}}{5^{a}} \cdot \frac{9^{a + 3}}{9^{a + 3}} - \frac{25^{a + 3}}{9^{a + 3}} \cdot \frac{5^{a}}{5^{a}}$

Step 4

Write each expression with a common denominator of

$5^{a} \cdot 9^{a + 3}$ , by multiplying each by an appropriate factor of

$1$ .

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Step 4.1

Multiply

$\frac{{\sqrt{15}}^{a}}{5^{a}}$ by

$\frac{9^{a + 3}}{9^{a + 3}}$ .

$\frac{{\sqrt{15}}^{a} \cdot 9^{a + 3}}{5^{a} \cdot 9^{a + 3}} - \frac{25^{a + 3}}{9^{a + 3}} \cdot \frac{5^{a}}{5^{a}}$

Step 4.2

Multiply

$\frac{25^{a + 3}}{9^{a + 3}}$ by

$\frac{5^{a}}{5^{a}}$ .

$\frac{{\sqrt{15}}^{a} \cdot 9^{a + 3}}{5^{a} \cdot 9^{a + 3}} - \frac{25^{a + 3} \cdot 5^{a}}{9^{a + 3} \cdot 5^{a}}$

Step 4.3

Reorder the factors of

$5^{a} \cdot 9^{a + 3}$ .

$\frac{{\sqrt{15}}^{a} \cdot 9^{a + 3}}{9^{a + 3} \cdot 5^{a}} - \frac{25^{a + 3} \cdot 5^{a}}{9^{a + 3} \cdot 5^{a}}$

Step 5

Combine the numerators over the common denominator.

$\frac{{\sqrt{15}}^{a} \cdot 9^{a + 3} - 25^{a + 3} \cdot 5^{a}}{9^{a + 3} \cdot 5^{a}}$

Step 6

Multiply

$- 25^{a + 3} \cdot 5^{a}$ .

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Step 6.1

Rewrite

$25$ as

$5^{2}$ .

$\frac{{\sqrt{15}}^{a} \cdot 9^{a + 3} - (5^{a} \cdot {(5^{2})}^{a + 3})}{9^{a + 3} \cdot 5^{a}}$

Step 6.2

Multiply the exponents in

${(5^{2})}^{a + 3}$ .

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Step 6.2.1

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{{\sqrt{15}}^{a} \cdot 9^{a + 3} - (5^{a} \cdot 5^{2 (a + 3)})}{9^{a + 3} \cdot 5^{a}}$

Step 6.2.2

Apply the distributive property.

$\frac{{\sqrt{15}}^{a} \cdot 9^{a + 3} - (5^{a} \cdot 5^{2 a + 2 \cdot 3})}{9^{a + 3} \cdot 5^{a}}$

Step 6.2.3

Multiply

$2$ by

$3$ .

$\frac{{\sqrt{15}}^{a} \cdot 9^{a + 3} - (5^{a} \cdot 5^{2 a + 6})}{9^{a + 3} \cdot 5^{a}}$

Step 6.3

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{\sqrt{15}}^{a} \cdot 9^{a + 3} - 5^{a + 2 a + 6}}{9^{a + 3} \cdot 5^{a}}$

Step 6.4

Add

$a$ and

$2 a$ .

$\frac{{\sqrt{15}}^{a} \cdot 9^{a + 3} - 5^{3 a + 6}}{9^{a + 3} \cdot 5^{a}}$