Basic Math Examples

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

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 $2 a$ .

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