Algebra Examples

\sqrt{15 n^{2}} \cdot \sqrt{10 n^{3}}

$\sqrt{15 n^{2}} \cdot \sqrt{10 n^{3}}$

Step 1

Reorder

15

$15$ and

n^{2}

$n^{2}$ .

\sqrt{n^{2} \cdot 15} \cdot \sqrt{10 n^{3}}

$\sqrt{n^{2} \cdot 15} \cdot \sqrt{10 n^{3}}$

Step 2

Pull terms out from under the radical.

| n | \sqrt{15} \cdot \sqrt{10 n^{3}}

$| n | \sqrt{15} \cdot \sqrt{10 n^{3}}$

Step 3

Rewrite

10 n^{3}

$10 n^{3}$ as

n^{2} \cdot (10 n)

$n^{2} \cdot (10 n)$ .

Tap for more steps...

Step 3.1

Factor out

n^{2}

$n^{2}$ .

| n | \sqrt{15} \cdot \sqrt{10 (n^{2} n)}

$| n | \sqrt{15} \cdot \sqrt{10 (n^{2} n)}$

Step 3.2

Reorder

10

$10$ and

n^{2}

$n^{2}$ .

| n | \sqrt{15} \cdot \sqrt{n^{2} \cdot 10 n}

$| n | \sqrt{15} \cdot \sqrt{n^{2} \cdot 10 n}$

Step 3.3

Add parentheses.

| n | \sqrt{15} \cdot \sqrt{n^{2} \cdot (10 n)}

$| n | \sqrt{15} \cdot \sqrt{n^{2} \cdot (10 n)}$

| n | \sqrt{15} \cdot \sqrt{n^{2} \cdot (10 n)}

$| n | \sqrt{15} \cdot \sqrt{n^{2} \cdot (10 n)}$

Step 4

Pull terms out from under the radical.

| n | \sqrt{15} \cdot (| n | \sqrt{10 n})

$| n | \sqrt{15} \cdot (| n | \sqrt{10 n})$

Step 5

Multiply

| n | \sqrt{15} (| n | \sqrt{10 n})

$| n | \sqrt{15} (| n | \sqrt{10 n})$ .

Tap for more steps...

Step 5.1

To multiply absolute values, multiply the terms inside each absolute value.

| n \cdot n | \sqrt{15} \sqrt{10 n}

$| n \cdot n | \sqrt{15} \sqrt{10 n}$

Step 5.2

Raise

n

$n$ to the power of

1

$1$ .

∣ ∣ n^{1} n ∣ ∣ \sqrt{15} \sqrt{10 n}

$| n^{1} n | \sqrt{15} \sqrt{10 n}$

Step 5.3

Raise

n

$n$ to the power of

1

$1$ .

∣ ∣ n^{1} n^{1} ∣ ∣ \sqrt{15} \sqrt{10 n}

$| n^{1} n^{1} | \sqrt{15} \sqrt{10 n}$

Step 5.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

∣ ∣ n^{1 + 1} ∣ ∣ \sqrt{15} \sqrt{10 n}

$| n^{1 + 1} | \sqrt{15} \sqrt{10 n}$

Step 5.5

Add

1

$1$ and

1

$1$ .

∣ ∣ n^{2} ∣ ∣ \sqrt{15} \sqrt{10 n}

$| n^{2} | \sqrt{15} \sqrt{10 n}$

Step 5.6

Combine using the product rule for radicals.

∣ ∣ n^{2} ∣ ∣ \sqrt{10 n \cdot 15}

$| n^{2} | \sqrt{10 n \cdot 15}$

Step 5.7

Multiply

15

$15$ by

10

$10$ .

∣ ∣ n^{2} ∣ ∣ \sqrt{150 n}

$| n^{2} | \sqrt{150 n}$

∣ ∣ n^{2} ∣ ∣ \sqrt{150 n}

$| n^{2} | \sqrt{150 n}$

Step 6

Remove non-negative terms from the absolute value.

n^{2} \sqrt{150 n}

$n^{2} \sqrt{150 n}$

Step 7

Rewrite

150 n

$150 n$ as

5^{2} \cdot (6 n)

$5^{2} \cdot (6 n)$ .

Tap for more steps...

Step 7.1

Factor

25

$25$ out of

150

$150$ .

n^{2} \sqrt{25 (6) n}

$n^{2} \sqrt{25 (6) n}$

Step 7.2

Rewrite

25

$25$ as

5^{2}

$5^{2}$ .

n^{2} \sqrt{5^{2} \cdot 6 n}

$n^{2} \sqrt{5^{2} \cdot 6 n}$

Step 7.3

Add parentheses.

n^{2} \sqrt{5^{2} \cdot (6 n)}

$n^{2} \sqrt{5^{2} \cdot (6 n)}$

n^{2} \sqrt{5^{2} \cdot (6 n)}

$n^{2} \sqrt{5^{2} \cdot (6 n)}$

Step 8

Pull terms out from under the radical.

n^{2} (| 5 | \sqrt{6 n})

$n^{2} (| 5 | \sqrt{6 n})$

Step 9

Rewrite using the commutative property of multiplication.

| 5 | n^{2} \sqrt{6 n}

$| 5 | n^{2} \sqrt{6 n}$

Step 10

The absolute value is the distance between a number and zero. The distance between

0

$0$ and

5

$5$ is

5

$5$ .

5 n^{2} \sqrt{6 n}

$5 n^{2} \sqrt{6 n}$