Basic Math Examples

${(\frac{- 3 \sqrt{10}}{10})}^{2} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1

Simplify each term.

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Step 1.1

Move the negative in front of the fraction.

${(- \frac{3 \sqrt{10}}{10})}^{2} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.2

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

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Step 1.2.1

Apply the product rule to $- \frac{3 \sqrt{10}}{10}$ .

${(- 1)}^{2} {(\frac{3 \sqrt{10}}{10})}^{2} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.2.2

Apply the product rule to $\frac{3 \sqrt{10}}{10}$ .

${(- 1)}^{2} \frac{{(3 \sqrt{10})}^{2}}{10^{2}} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.2.3

Apply the product rule to $3 \sqrt{10}$ .

${(- 1)}^{2} \frac{3^{2} {\sqrt{10}}^{2}}{10^{2}} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.3

Raise $- 1$ to the power of $2$ .

$1 \frac{3^{2} {\sqrt{10}}^{2}}{10^{2}} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.4

Multiply $\frac{3^{2} {\sqrt{10}}^{2}}{10^{2}}$ by $1$ .

$\frac{3^{2} {\sqrt{10}}^{2}}{10^{2}} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Raise $3$ to the power of $2$ .

$\frac{9 {\sqrt{10}}^{2}}{10^{2}} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.5.2

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 1.5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{9 {(10^{\frac{1}{2}})}^{2}}{10^{2}} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.5.2.2

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

$\frac{9 \cdot 10^{\frac{1}{2} \cdot 2}}{10^{2}} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.5.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{9 \cdot 10^{\frac{2}{2}}}{10^{2}} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.5.2.4

Cancel the common factor of $2$ .

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Step 1.5.2.4.1

Cancel the common factor.

$\frac{9 \cdot 10^{\frac{2}{2}}}{10^{2}} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.5.2.4.2

Rewrite the expression.

$\frac{9 \cdot 10^{1}}{10^{2}} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.5.2.5

Evaluate the exponent.

$\frac{9 \cdot 10}{10^{2}} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.6

Raise $10$ to the power of $2$ .

$\frac{9 \cdot 10}{100} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.7

Multiply $9$ by $10$ .

$\frac{90}{100} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.8

Cancel the common factor of $90$ and $100$ .

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Step 1.8.1

Factor $10$ out of $90$ .

$\frac{10 (9)}{100} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.8.2

Cancel the common factors.

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Step 1.8.2.1

Factor $10$ out of $100$ .

$\frac{10 \cdot 9}{10 \cdot 10} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.8.2.2

Cancel the common factor.

$\frac{10 \cdot 9}{10 \cdot 10} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.8.2.3

Rewrite the expression.

$\frac{9}{10} - {(- \frac{\sqrt{10}}{10})}^{2}$

Step 1.9

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

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Step 1.9.1

Apply the product rule to $- \frac{\sqrt{10}}{10}$ .

$\frac{9}{10} - ({(- 1)}^{2} {(\frac{\sqrt{10}}{10})}^{2})$

Step 1.9.2

Apply the product rule to $\frac{\sqrt{10}}{10}$ .

$\frac{9}{10} - ({(- 1)}^{2} \frac{{\sqrt{10}}^{2}}{10^{2}})$

Step 1.10

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.10.1

Move ${(- 1)}^{2}$ .

$\frac{9}{10} + {(- 1)}^{2} \cdot - 1 \frac{{\sqrt{10}}^{2}}{10^{2}}$

Step 1.10.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 1.10.2.1

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

$\frac{9}{10} + {(- 1)}^{2} \cdot {(- 1)}^{1} \frac{{\sqrt{10}}^{2}}{10^{2}}$

Step 1.10.2.2

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

$\frac{9}{10} + {(- 1)}^{2 + 1} \frac{{\sqrt{10}}^{2}}{10^{2}}$

Step 1.10.3

Add $2$ and $1$ .

$\frac{9}{10} + {(- 1)}^{3} \frac{{\sqrt{10}}^{2}}{10^{2}}$

Step 1.11

Raise $- 1$ to the power of $3$ .

$\frac{9}{10} - \frac{{\sqrt{10}}^{2}}{10^{2}}$

Step 1.12

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 1.12.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\frac{9}{10} - \frac{{(10^{\frac{1}{2}})}^{2}}{10^{2}}$

Step 1.12.2

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

$\frac{9}{10} - \frac{10^{\frac{1}{2} \cdot 2}}{10^{2}}$

Step 1.12.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{9}{10} - \frac{10^{\frac{2}{2}}}{10^{2}}$

Step 1.12.4

Cancel the common factor of $2$ .

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Step 1.12.4.1

Cancel the common factor.

$\frac{9}{10} - \frac{10^{\frac{2}{2}}}{10^{2}}$

Step 1.12.4.2

Rewrite the expression.

$\frac{9}{10} - \frac{10^{1}}{10^{2}}$

Step 1.12.5

Evaluate the exponent.

$\frac{9}{10} - \frac{10}{10^{2}}$

Step 1.13

Raise $10$ to the power of $2$ .

$\frac{9}{10} - \frac{10}{100}$

Step 1.14

Cancel the common factor of $10$ and $100$ .

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Step 1.14.1

Factor $10$ out of $10$ .

$\frac{9}{10} - \frac{10 (1)}{100}$

Step 1.14.2

Cancel the common factors.

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Step 1.14.2.1

Factor $10$ out of $100$ .

$\frac{9}{10} - \frac{10 \cdot 1}{10 \cdot 10}$

Step 1.14.2.2

Cancel the common factor.

$\frac{9}{10} - \frac{10 \cdot 1}{10 \cdot 10}$

Step 1.14.2.3

Rewrite the expression.

$\frac{9}{10} - \frac{1}{10}$

Step 2

Simplify terms.

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Step 2.1

Combine the numerators over the common denominator.

$\frac{9 - 1}{10}$

Step 2.2

Subtract $1$ from $9$ .

$\frac{8}{10}$

Step 2.3

Cancel the common factor of $8$ and $10$ .

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Step 2.3.1

Factor $2$ out of $8$ .

$\frac{2 (4)}{10}$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 4}{2 \cdot 5}$

Step 2.3.2.2

Cancel the common factor.

$\frac{2 \cdot 4}{2 \cdot 5}$

Step 2.3.2.3

Rewrite the expression.

$\frac{4}{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{4}{5}$

Decimal Form:

$0.8$