Pre-Algebra Examples

\sqrt{144 {(x + 9)}^{10}}

$\sqrt{144 {(x + 9)}^{10}}$

Step 1

Rewrite

144 {(x + 9)}^{10}

$144 {(x + 9)}^{10}$ as

{(12 {(x + 9)}^{5})}^{2}

${(12 {(x + 9)}^{5})}^{2}$ .

\sqrt{{(12 {(x + 9)}^{5})}^{2}}

$\sqrt{{(12 {(x + 9)}^{5})}^{2}}$

Step 2

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

12 {(x + 9)}^{5}

$12 {(x + 9)}^{5}$

Step 3

Use the Binomial Theorem.

12 (x^{5} + 5 x^{4} \cdot 9 + 10 x^{3} \cdot 9^{2} + 10 x^{2} \cdot 9^{3} + 5 x \cdot 9^{4} + 9^{5})

$12 (x^{5} + 5 x^{4} \cdot 9 + 10 x^{3} \cdot 9^{2} + 10 x^{2} \cdot 9^{3} + 5 x \cdot 9^{4} + 9^{5})$

Step 4

Simplify terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply

9

$9$ by

5

$5$ .

12 (x^{5} + 45 x^{4} + 10 x^{3} \cdot 9^{2} + 10 x^{2} \cdot 9^{3} + 5 x \cdot 9^{4} + 9^{5})

$12 (x^{5} + 45 x^{4} + 10 x^{3} \cdot 9^{2} + 10 x^{2} \cdot 9^{3} + 5 x \cdot 9^{4} + 9^{5})$

Step 4.1.2

Raise

9

$9$ to the power of

2

$2$ .

12 (x^{5} + 45 x^{4} + 10 x^{3} \cdot 81 + 10 x^{2} \cdot 9^{3} + 5 x \cdot 9^{4} + 9^{5})

$12 (x^{5} + 45 x^{4} + 10 x^{3} \cdot 81 + 10 x^{2} \cdot 9^{3} + 5 x \cdot 9^{4} + 9^{5})$

Step 4.1.3

Multiply

81

$81$ by

10

$10$ .

12 (x^{5} + 45 x^{4} + 810 x^{3} + 10 x^{2} \cdot 9^{3} + 5 x \cdot 9^{4} + 9^{5})

$12 (x^{5} + 45 x^{4} + 810 x^{3} + 10 x^{2} \cdot 9^{3} + 5 x \cdot 9^{4} + 9^{5})$

Step 4.1.4

Raise

9

$9$ to the power of

3

$3$ .

12 (x^{5} + 45 x^{4} + 810 x^{3} + 10 x^{2} \cdot 729 + 5 x \cdot 9^{4} + 9^{5})

$12 (x^{5} + 45 x^{4} + 810 x^{3} + 10 x^{2} \cdot 729 + 5 x \cdot 9^{4} + 9^{5})$

Step 4.1.5

Multiply

729

$729$ by

10

$10$ .

12 (x^{5} + 45 x^{4} + 810 x^{3} + 7290 x^{2} + 5 x \cdot 9^{4} + 9^{5})

$12 (x^{5} + 45 x^{4} + 810 x^{3} + 7290 x^{2} + 5 x \cdot 9^{4} + 9^{5})$

Step 4.1.6

Raise

9

$9$ to the power of

4

$4$ .

12 (x^{5} + 45 x^{4} + 810 x^{3} + 7290 x^{2} + 5 x \cdot 6561 + 9^{5})

$12 (x^{5} + 45 x^{4} + 810 x^{3} + 7290 x^{2} + 5 x \cdot 6561 + 9^{5})$

Step 4.1.7

Multiply

6561

$6561$ by

5

$5$ .

12 (x^{5} + 45 x^{4} + 810 x^{3} + 7290 x^{2} + 32805 x + 9^{5})

$12 (x^{5} + 45 x^{4} + 810 x^{3} + 7290 x^{2} + 32805 x + 9^{5})$

Step 4.1.8

Raise

9

$9$ to the power of

5

$5$ .

12 (x^{5} + 45 x^{4} + 810 x^{3} + 7290 x^{2} + 32805 x + 59049)

$12 (x^{5} + 45 x^{4} + 810 x^{3} + 7290 x^{2} + 32805 x + 59049)$

12 (x^{5} + 45 x^{4} + 810 x^{3} + 7290 x^{2} + 32805 x + 59049)

$12 (x^{5} + 45 x^{4} + 810 x^{3} + 7290 x^{2} + 32805 x + 59049)$

Step 4.2

Apply the distributive property.

12 x^{5} + 12 (45 x^{4}) + 12 (810 x^{3}) + 12 (7290 x^{2}) + 12 (32805 x) + 12 \cdot 59049

$12 x^{5} + 12 (45 x^{4}) + 12 (810 x^{3}) + 12 (7290 x^{2}) + 12 (32805 x) + 12 \cdot 59049$

12 x^{5} + 12 (45 x^{4}) + 12 (810 x^{3}) + 12 (7290 x^{2}) + 12 (32805 x) + 12 \cdot 59049

$12 x^{5} + 12 (45 x^{4}) + 12 (810 x^{3}) + 12 (7290 x^{2}) + 12 (32805 x) + 12 \cdot 59049$

Step 5

Simplify.

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Step 5.1

Multiply

45

$45$ by

12

$12$ .

12 x^{5} + 540 x^{4} + 12 (810 x^{3}) + 12 (7290 x^{2}) + 12 (32805 x) + 12 \cdot 59049

$12 x^{5} + 540 x^{4} + 12 (810 x^{3}) + 12 (7290 x^{2}) + 12 (32805 x) + 12 \cdot 59049$

Step 5.2

Multiply

810

$810$ by

12

$12$ .

12 x^{5} + 540 x^{4} + 9720 x^{3} + 12 (7290 x^{2}) + 12 (32805 x) + 12 \cdot 59049

$12 x^{5} + 540 x^{4} + 9720 x^{3} + 12 (7290 x^{2}) + 12 (32805 x) + 12 \cdot 59049$

Step 5.3

Multiply

7290

$7290$ by

12

$12$ .

12 x^{5} + 540 x^{4} + 9720 x^{3} + 87480 x^{2} + 12 (32805 x) + 12 \cdot 59049

$12 x^{5} + 540 x^{4} + 9720 x^{3} + 87480 x^{2} + 12 (32805 x) + 12 \cdot 59049$

Step 5.4

Multiply

32805

$32805$ by

12

$12$ .

12 x^{5} + 540 x^{4} + 9720 x^{3} + 87480 x^{2} + 393660 x + 12 \cdot 59049

$12 x^{5} + 540 x^{4} + 9720 x^{3} + 87480 x^{2} + 393660 x + 12 \cdot 59049$

Step 5.5

Multiply

12

$12$ by

59049

$59049$ .

12 x^{5} + 540 x^{4} + 9720 x^{3} + 87480 x^{2} + 393660 x + 708588

$12 x^{5} + 540 x^{4} + 9720 x^{3} + 87480 x^{2} + 393660 x + 708588$

12 x^{5} + 540 x^{4} + 9720 x^{3} + 87480 x^{2} + 393660 x + 708588

$12 x^{5} + 540 x^{4} + 9720 x^{3} + 87480 x^{2} + 393660 x + 708588$