Algebra Examples

${(x^{2} + 5 x + 25)}^{\frac{3}{2}} = 343$

Step 1

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${({(x^{2} + 5 x + 25)}^{\frac{3}{2}})}^{\frac{2}{3}} = 343^{\frac{2}{3}}$

Step 2

Simplify the exponent.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify ${({(x^{2} + 5 x + 25)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 2.1.1.1

Multiply the exponents in ${({(x^{2} + 5 x + 25)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 2.1.1.1.1

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

${(x^{2} + 5 x + 25)}^{\frac{3}{2} \cdot \frac{2}{3}} = 343^{\frac{2}{3}}$

Step 2.1.1.1.2

Cancel the common factor of $3$ .

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Step 2.1.1.1.2.1

Cancel the common factor.

${(x^{2} + 5 x + 25)}^{\frac{3}{2} \cdot \frac{2}{3}} = 343^{\frac{2}{3}}$

Step 2.1.1.1.2.2

Rewrite the expression.

${(x^{2} + 5 x + 25)}^{\frac{1}{2} \cdot 2} = 343^{\frac{2}{3}}$

Step 2.1.1.1.3

Cancel the common factor of $2$ .

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Step 2.1.1.1.3.1

Cancel the common factor.

${(x^{2} + 5 x + 25)}^{\frac{1}{2} \cdot 2} = 343^{\frac{2}{3}}$

Step 2.1.1.1.3.2

Rewrite the expression.

${(x^{2} + 5 x + 25)}^{1} = 343^{\frac{2}{3}}$

Step 2.1.1.2

Simplify.

$x^{2} + 5 x + 25 = 343^{\frac{2}{3}}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $343^{\frac{2}{3}}$ .

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Step 2.2.1.1

Simplify the expression.

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Step 2.2.1.1.1

Rewrite $343$ as $7^{3}$ .

$x^{2} + 5 x + 25 = {(7^{3})}^{\frac{2}{3}}$

Step 2.2.1.1.2

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

$x^{2} + 5 x + 25 = 7^{3 (\frac{2}{3})}$

Step 2.2.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.2.1

Cancel the common factor.

$x^{2} + 5 x + 25 = 7^{3 (\frac{2}{3})}$

Step 2.2.1.2.2

Rewrite the expression.

$x^{2} + 5 x + 25 = 7^{2}$

Step 2.2.1.3

Raise $7$ to the power of $2$ .

$x^{2} + 5 x + 25 = 49$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $49$ from both sides of the equation.

$x^{2} + 5 x + 25 - 49 = 0$

Step 3.2

Subtract $49$ from $25$ .

$x^{2} + 5 x - 24 = 0$

Step 3.3

Factor $x^{2} + 5 x - 24$ using the AC method.

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Step 3.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 24$ and whose sum is $5$ .

$- 3, 8$

Step 3.3.2

Write the factored form using these integers.

$(x - 3) (x + 8) = 0$

Step 3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 3 = 0$

$x + 8 = 0$

Step 3.5

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 3.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 3.6

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 3.6.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 3.7

The final solution is all the values that make $(x - 3) (x + 8) = 0$ true.

$x = 3, - 8$