Simplify (5 square root of 2-1)^2

${(5 \sqrt{2} - 1)}^{2}$

Step 1

Rewrite ${(5 \sqrt{2} - 1)}^{2}$ as $(5 \sqrt{2} - 1) (5 \sqrt{2} - 1)$ .

$(5 \sqrt{2} - 1) (5 \sqrt{2} - 1)$

Step 2

Expand $(5 \sqrt{2} - 1) (5 \sqrt{2} - 1)$ using the FOIL Method.

Apply the distributive property.

$5 \sqrt{2} (5 \sqrt{2} - 1) - 1 (5 \sqrt{2} - 1)$

Apply the distributive property.

$5 \sqrt{2} (5 \sqrt{2}) + 5 \sqrt{2} \cdot - 1 - 1 (5 \sqrt{2} - 1)$

Apply the distributive property.

$5 \sqrt{2} (5 \sqrt{2}) + 5 \sqrt{2} \cdot - 1 - 1 (5 \sqrt{2}) - 1 \cdot - 1$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $5 \sqrt{2} (5 \sqrt{2})$ .

Multiply $5$ by $5$ .

$25 \sqrt{2} \sqrt{2} + 5 \sqrt{2} \cdot - 1 - 1 (5 \sqrt{2}) - 1 \cdot - 1$

Raise $\sqrt{2}$ to the power of $1$ .

$25 ({\sqrt{2}}^{1} \sqrt{2}) + 5 \sqrt{2} \cdot - 1 - 1 (5 \sqrt{2}) - 1 \cdot - 1$

Raise $\sqrt{2}$ to the power of $1$ .

$25 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 5 \sqrt{2} \cdot - 1 - 1 (5 \sqrt{2}) - 1 \cdot - 1$

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

$25 {\sqrt{2}}^{1 + 1} + 5 \sqrt{2} \cdot - 1 - 1 (5 \sqrt{2}) - 1 \cdot - 1$

Add $1$ and $1$ .

$25 {\sqrt{2}}^{2} + 5 \sqrt{2} \cdot - 1 - 1 (5 \sqrt{2}) - 1 \cdot - 1$

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

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

$25 {(2^{\frac{1}{2}})}^{2} + 5 \sqrt{2} \cdot - 1 - 1 (5 \sqrt{2}) - 1 \cdot - 1$

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

$25 \cdot 2^{\frac{1}{2} \cdot 2} + 5 \sqrt{2} \cdot - 1 - 1 (5 \sqrt{2}) - 1 \cdot - 1$

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

$25 \cdot 2^{\frac{2}{2}} + 5 \sqrt{2} \cdot - 1 - 1 (5 \sqrt{2}) - 1 \cdot - 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$25 \cdot 2^{\frac{2}{2}} + 5 \sqrt{2} \cdot - 1 - 1 (5 \sqrt{2}) - 1 \cdot - 1$

Rewrite the expression.

$25 \cdot 2^{1} + 5 \sqrt{2} \cdot - 1 - 1 (5 \sqrt{2}) - 1 \cdot - 1$

Evaluate the exponent.

$25 \cdot 2 + 5 \sqrt{2} \cdot - 1 - 1 (5 \sqrt{2}) - 1 \cdot - 1$

Multiply $25$ by $2$ .

$50 + 5 \sqrt{2} \cdot - 1 - 1 (5 \sqrt{2}) - 1 \cdot - 1$

Multiply $- 1$ by $5$ .

$50 - 5 \sqrt{2} - 1 (5 \sqrt{2}) - 1 \cdot - 1$

Multiply $5$ by $- 1$ .

$50 - 5 \sqrt{2} - 5 \sqrt{2} - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$50 - 5 \sqrt{2} - 5 \sqrt{2} + 1$

Add $50$ and $1$ .

$51 - 5 \sqrt{2} - 5 \sqrt{2}$

Subtract $5 \sqrt{2}$ from $- 5 \sqrt{2}$ .

$51 - 10 \sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$51 - 10 \sqrt{2}$

Decimal Form:

$36.85786437 \dots$

Simplify (5 square root of 2-1)^2

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