Algebra Examples

$\sqrt{{(a^{2} + 4 a)}^{12}}$

Step 1

Factor $a$ out of $a^{2} + 4 a$ .

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Step 1.1

Factor $a$ out of $a^{2}$ .

$\sqrt{{(a \cdot a + 4 a)}^{12}}$

Step 1.2

Factor $a$ out of $4 a$ .

$\sqrt{{(a \cdot a + a \cdot 4)}^{12}}$

Step 1.3

Factor $a$ out of $a \cdot a + a \cdot 4$ .

$\sqrt{{(a (a + 4))}^{12}}$

Step 2

Apply the product rule to $a (a + 4)$ .

$\sqrt{a^{12} {(a + 4)}^{12}}$

Step 3

Rewrite $a^{12} {(a + 4)}^{12}$ as ${(a^{6} {(a + 4)}^{6})}^{2}$ .

$\sqrt{{(a^{6} {(a + 4)}^{6})}^{2}}$

Step 4

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

$a^{6} {(a + 4)}^{6}$

Step 5

Use the Binomial Theorem.

$a^{6} (a^{6} + 6 a^{5} \cdot 4 + 15 a^{4} \cdot 4^{2} + 20 a^{3} \cdot 4^{3} + 15 a^{2} \cdot 4^{4} + 6 a \cdot 4^{5} + 4^{6})$

Step 6

Simplify terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $4$ by $6$ .

$a^{6} (a^{6} + 24 a^{5} + 15 a^{4} \cdot 4^{2} + 20 a^{3} \cdot 4^{3} + 15 a^{2} \cdot 4^{4} + 6 a \cdot 4^{5} + 4^{6})$

Step 6.1.2

Raise $4$ to the power of $2$ .

$a^{6} (a^{6} + 24 a^{5} + 15 a^{4} \cdot 16 + 20 a^{3} \cdot 4^{3} + 15 a^{2} \cdot 4^{4} + 6 a \cdot 4^{5} + 4^{6})$

Step 6.1.3

Multiply $16$ by $15$ .

$a^{6} (a^{6} + 24 a^{5} + 240 a^{4} + 20 a^{3} \cdot 4^{3} + 15 a^{2} \cdot 4^{4} + 6 a \cdot 4^{5} + 4^{6})$

Step 6.1.4

Raise $4$ to the power of $3$ .

$a^{6} (a^{6} + 24 a^{5} + 240 a^{4} + 20 a^{3} \cdot 64 + 15 a^{2} \cdot 4^{4} + 6 a \cdot 4^{5} + 4^{6})$

Step 6.1.5

Multiply $64$ by $20$ .

$a^{6} (a^{6} + 24 a^{5} + 240 a^{4} + 1280 a^{3} + 15 a^{2} \cdot 4^{4} + 6 a \cdot 4^{5} + 4^{6})$

Step 6.1.6

Raise $4$ to the power of $4$ .

$a^{6} (a^{6} + 24 a^{5} + 240 a^{4} + 1280 a^{3} + 15 a^{2} \cdot 256 + 6 a \cdot 4^{5} + 4^{6})$

Step 6.1.7

Multiply $256$ by $15$ .

$a^{6} (a^{6} + 24 a^{5} + 240 a^{4} + 1280 a^{3} + 3840 a^{2} + 6 a \cdot 4^{5} + 4^{6})$

Step 6.1.8

Raise $4$ to the power of $5$ .

$a^{6} (a^{6} + 24 a^{5} + 240 a^{4} + 1280 a^{3} + 3840 a^{2} + 6 a \cdot 1024 + 4^{6})$

Step 6.1.9

Multiply $1024$ by $6$ .

$a^{6} (a^{6} + 24 a^{5} + 240 a^{4} + 1280 a^{3} + 3840 a^{2} + 6144 a + 4^{6})$

Step 6.1.10

Raise $4$ to the power of $6$ .

$a^{6} (a^{6} + 24 a^{5} + 240 a^{4} + 1280 a^{3} + 3840 a^{2} + 6144 a + 4096)$

Step 6.2

Apply the distributive property.

$a^{6} a^{6} + a^{6} (24 a^{5}) + a^{6} (240 a^{4}) + a^{6} (1280 a^{3}) + a^{6} (3840 a^{2}) + a^{6} (6144 a) + a^{6} \cdot 4096$

Step 7

Simplify.

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Step 7.1

Multiply $a^{6}$ by $a^{6}$ by adding the exponents.

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Step 7.1.1

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

$a^{6 + 6} + a^{6} (24 a^{5}) + a^{6} (240 a^{4}) + a^{6} (1280 a^{3}) + a^{6} (3840 a^{2}) + a^{6} (6144 a) + a^{6} \cdot 4096$

Step 7.1.2

Add $6$ and $6$ .

$a^{12} + a^{6} (24 a^{5}) + a^{6} (240 a^{4}) + a^{6} (1280 a^{3}) + a^{6} (3840 a^{2}) + a^{6} (6144 a) + a^{6} \cdot 4096$

Step 7.2

Rewrite using the commutative property of multiplication.

$a^{12} + 24 a^{6} a^{5} + a^{6} (240 a^{4}) + a^{6} (1280 a^{3}) + a^{6} (3840 a^{2}) + a^{6} (6144 a) + a^{6} \cdot 4096$

Step 7.3

Rewrite using the commutative property of multiplication.

$a^{12} + 24 a^{6} a^{5} + 240 a^{6} a^{4} + a^{6} (1280 a^{3}) + a^{6} (3840 a^{2}) + a^{6} (6144 a) + a^{6} \cdot 4096$

Step 7.4

Rewrite using the commutative property of multiplication.

$a^{12} + 24 a^{6} a^{5} + 240 a^{6} a^{4} + 1280 a^{6} a^{3} + a^{6} (3840 a^{2}) + a^{6} (6144 a) + a^{6} \cdot 4096$

Step 7.5

Rewrite using the commutative property of multiplication.

$a^{12} + 24 a^{6} a^{5} + 240 a^{6} a^{4} + 1280 a^{6} a^{3} + 3840 a^{6} a^{2} + a^{6} (6144 a) + a^{6} \cdot 4096$

Step 7.6

Rewrite using the commutative property of multiplication.

$a^{12} + 24 a^{6} a^{5} + 240 a^{6} a^{4} + 1280 a^{6} a^{3} + 3840 a^{6} a^{2} + 6144 a^{6} a + a^{6} \cdot 4096$

Step 7.7

Move $4096$ to the left of $a^{6}$ .

$a^{12} + 24 a^{6} a^{5} + 240 a^{6} a^{4} + 1280 a^{6} a^{3} + 3840 a^{6} a^{2} + 6144 a^{6} a + 4096 \cdot a^{6}$

Step 8

Simplify each term.

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Step 8.1

Multiply $a^{6}$ by $a^{5}$ by adding the exponents.

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Step 8.1.1

Move $a^{5}$ .

$a^{12} + 24 (a^{5} a^{6}) + 240 a^{6} a^{4} + 1280 a^{6} a^{3} + 3840 a^{6} a^{2} + 6144 a^{6} a + 4096 \cdot a^{6}$

Step 8.1.2

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

$a^{12} + 24 a^{5 + 6} + 240 a^{6} a^{4} + 1280 a^{6} a^{3} + 3840 a^{6} a^{2} + 6144 a^{6} a + 4096 \cdot a^{6}$

Step 8.1.3

Add $5$ and $6$ .

$a^{12} + 24 a^{11} + 240 a^{6} a^{4} + 1280 a^{6} a^{3} + 3840 a^{6} a^{2} + 6144 a^{6} a + 4096 \cdot a^{6}$

Step 8.2

Multiply $a^{6}$ by $a^{4}$ by adding the exponents.

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Step 8.2.1

Move $a^{4}$ .

$a^{12} + 24 a^{11} + 240 (a^{4} a^{6}) + 1280 a^{6} a^{3} + 3840 a^{6} a^{2} + 6144 a^{6} a + 4096 \cdot a^{6}$

Step 8.2.2

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

$a^{12} + 24 a^{11} + 240 a^{4 + 6} + 1280 a^{6} a^{3} + 3840 a^{6} a^{2} + 6144 a^{6} a + 4096 \cdot a^{6}$

Step 8.2.3

Add $4$ and $6$ .

$a^{12} + 24 a^{11} + 240 a^{10} + 1280 a^{6} a^{3} + 3840 a^{6} a^{2} + 6144 a^{6} a + 4096 \cdot a^{6}$

Step 8.3

Multiply $a^{6}$ by $a^{3}$ by adding the exponents.

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Step 8.3.1

Move $a^{3}$ .

$a^{12} + 24 a^{11} + 240 a^{10} + 1280 (a^{3} a^{6}) + 3840 a^{6} a^{2} + 6144 a^{6} a + 4096 \cdot a^{6}$

Step 8.3.2

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

$a^{12} + 24 a^{11} + 240 a^{10} + 1280 a^{3 + 6} + 3840 a^{6} a^{2} + 6144 a^{6} a + 4096 \cdot a^{6}$

Step 8.3.3

Add $3$ and $6$ .

$a^{12} + 24 a^{11} + 240 a^{10} + 1280 a^{9} + 3840 a^{6} a^{2} + 6144 a^{6} a + 4096 \cdot a^{6}$

Step 8.4

Multiply $a^{6}$ by $a^{2}$ by adding the exponents.

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Step 8.4.1

Move $a^{2}$ .

$a^{12} + 24 a^{11} + 240 a^{10} + 1280 a^{9} + 3840 (a^{2} a^{6}) + 6144 a^{6} a + 4096 \cdot a^{6}$

Step 8.4.2

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

$a^{12} + 24 a^{11} + 240 a^{10} + 1280 a^{9} + 3840 a^{2 + 6} + 6144 a^{6} a + 4096 \cdot a^{6}$

Step 8.4.3

Add $2$ and $6$ .

$a^{12} + 24 a^{11} + 240 a^{10} + 1280 a^{9} + 3840 a^{8} + 6144 a^{6} a + 4096 \cdot a^{6}$

Step 8.5

Multiply $a^{6}$ by $a$ by adding the exponents.

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Step 8.5.1

Move $a$ .

$a^{12} + 24 a^{11} + 240 a^{10} + 1280 a^{9} + 3840 a^{8} + 6144 (a \cdot a^{6}) + 4096 \cdot a^{6}$

Step 8.5.2

Multiply $a$ by $a^{6}$ .

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Step 8.5.2.1

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

$a^{12} + 24 a^{11} + 240 a^{10} + 1280 a^{9} + 3840 a^{8} + 6144 (a^{1} a^{6}) + 4096 \cdot a^{6}$

Step 8.5.2.2

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

$a^{12} + 24 a^{11} + 240 a^{10} + 1280 a^{9} + 3840 a^{8} + 6144 a^{1 + 6} + 4096 \cdot a^{6}$

Step 8.5.3

Add $1$ and $6$ .

$a^{12} + 24 a^{11} + 240 a^{10} + 1280 a^{9} + 3840 a^{8} + 6144 a^{7} + 4096 \cdot a^{6}$

$a^{12} + 24 a^{11} + 240 a^{10} + 1280 a^{9} + 3840 a^{8} + 6144 a^{7} + 4096 a^{6}$