Algebra Examples

$(\sqrt{5} + \sqrt{6} + \sqrt{7}) \cdot (\sqrt{5} + \sqrt{6} - \sqrt{7})$

Step 1

Expand $(\sqrt{5} + \sqrt{6} + \sqrt{7}) (\sqrt{5} + \sqrt{6} - \sqrt{7})$ by multiplying each term in the first expression by each term in the second expression.

$\sqrt{5} \sqrt{5} + \sqrt{5} \sqrt{6} + \sqrt{5} (- \sqrt{7}) + \sqrt{6} \sqrt{5} + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{7}) + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Combine using the product rule for radicals.

$\sqrt{5 \cdot 5} + \sqrt{5} \sqrt{6} + \sqrt{5} (- \sqrt{7}) + \sqrt{6} \sqrt{5} + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{7}) + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.2

Multiply $5$ by $5$ .

$\sqrt{25} + \sqrt{5} \sqrt{6} + \sqrt{5} (- \sqrt{7}) + \sqrt{6} \sqrt{5} + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{7}) + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.3

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2}} + \sqrt{5} \sqrt{6} + \sqrt{5} (- \sqrt{7}) + \sqrt{6} \sqrt{5} + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{7}) + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.4

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

$5 + \sqrt{5} \sqrt{6} + \sqrt{5} (- \sqrt{7}) + \sqrt{6} \sqrt{5} + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{7}) + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.5

Combine using the product rule for radicals.

$5 + \sqrt{5 \cdot 6} + \sqrt{5} (- \sqrt{7}) + \sqrt{6} \sqrt{5} + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{7}) + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.6

Multiply $5$ by $6$ .

$5 + \sqrt{30} + \sqrt{5} (- \sqrt{7}) + \sqrt{6} \sqrt{5} + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{7}) + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.7

Multiply $\sqrt{5} (- \sqrt{7})$ .

Tap for more steps...

Step 2.1.7.1

Combine using the product rule for radicals.

$5 + \sqrt{30} - \sqrt{5 \cdot 7} + \sqrt{6} \sqrt{5} + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{7}) + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.7.2

Multiply $5$ by $7$ .

$5 + \sqrt{30} - \sqrt{35} + \sqrt{6} \sqrt{5} + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{7}) + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.8

Combine using the product rule for radicals.

$5 + \sqrt{30} - \sqrt{35} + \sqrt{6 \cdot 5} + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{7}) + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.9

Multiply $6$ by $5$ .

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + \sqrt{6} \sqrt{6} + \sqrt{6} (- \sqrt{7}) + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.10

Combine using the product rule for radicals.

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + \sqrt{6 \cdot 6} + \sqrt{6} (- \sqrt{7}) + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.11

Multiply $6$ by $6$ .

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + \sqrt{36} + \sqrt{6} (- \sqrt{7}) + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.12

Rewrite $36$ as $6^{2}$ .

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + \sqrt{6^{2}} + \sqrt{6} (- \sqrt{7}) + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.13

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

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 + \sqrt{6} (- \sqrt{7}) + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.14

Multiply $\sqrt{6} (- \sqrt{7})$ .

Tap for more steps...

Step 2.1.14.1

Combine using the product rule for radicals.

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{6 \cdot 7} + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.14.2

Multiply $6$ by $7$ .

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{7} \sqrt{5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.15

Combine using the product rule for radicals.

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{7 \cdot 5} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.16

Multiply $7$ by $5$ .

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{35} + \sqrt{7} \sqrt{6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.17

Combine using the product rule for radicals.

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{35} + \sqrt{7 \cdot 6} + \sqrt{7} (- \sqrt{7})$

Step 2.1.18

Multiply $7$ by $6$ .

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{35} + \sqrt{42} + \sqrt{7} (- \sqrt{7})$

Step 2.1.19

Multiply $\sqrt{7} (- \sqrt{7})$ .

Tap for more steps...

Step 2.1.19.1

Raise $\sqrt{7}$ to the power of $1$ .

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{35} + \sqrt{42} - ({\sqrt{7}}^{1} \sqrt{7})$

Step 2.1.19.2

Raise $\sqrt{7}$ to the power of $1$ .

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{35} + \sqrt{42} - ({\sqrt{7}}^{1} {\sqrt{7}}^{1})$

Step 2.1.19.3

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

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{35} + \sqrt{42} - {\sqrt{7}}^{1 + 1}$

Step 2.1.19.4

Add $1$ and $1$ .

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{35} + \sqrt{42} - {\sqrt{7}}^{2}$

Step 2.1.20

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Tap for more steps...

Step 2.1.20.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{35} + \sqrt{42} - {(7^{\frac{1}{2}})}^{2}$

Step 2.1.20.2

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

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{35} + \sqrt{42} - 7^{\frac{1}{2} \cdot 2}$

Step 2.1.20.3

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

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{35} + \sqrt{42} - 7^{\frac{2}{2}}$

Step 2.1.20.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.20.4.1

Cancel the common factor.

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{35} + \sqrt{42} - 7^{\frac{2}{2}}$

Step 2.1.20.4.2

Rewrite the expression.

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{35} + \sqrt{42} - 7^{1}$

Step 2.1.20.5

Evaluate the exponent.

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{35} + \sqrt{42} - 1 \cdot 7$

Step 2.1.21

Multiply $- 1$ by $7$ .

$5 + \sqrt{30} - \sqrt{35} + \sqrt{30} + 6 - \sqrt{42} + \sqrt{35} + \sqrt{42} - 7$

Step 2.2

Simplify by adding terms.

Tap for more steps...

Step 2.2.1

Add $5$ and $6$ .

$11 + \sqrt{30} - \sqrt{35} + \sqrt{30} - \sqrt{42} + \sqrt{35} + \sqrt{42} - 7$

Step 2.2.2

Subtract $7$ from $11$ .

$4 + \sqrt{30} - \sqrt{35} + \sqrt{30} - \sqrt{42} + \sqrt{35} + \sqrt{42}$

Step 2.2.3

Add $\sqrt{30}$ and $\sqrt{30}$ .

$4 + 2 \sqrt{30} - \sqrt{35} - \sqrt{42} + \sqrt{35} + \sqrt{42}$

Step 2.2.4

Add $- \sqrt{35}$ and $\sqrt{35}$ .

$4 + 2 \sqrt{30} + 0 - \sqrt{42} + \sqrt{42}$

Step 2.2.5

Add $4$ and $0$ .

$4 + 2 \sqrt{30} - \sqrt{42} + \sqrt{42}$

Step 2.2.6

Add $- \sqrt{42}$ and $\sqrt{42}$ .

$4 + 2 \sqrt{30} + 0$

Step 2.2.7

Add $4$ and $0$ .

$4 + 2 \sqrt{30}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$4 + 2 \sqrt{30}$

Decimal Form:

$14.95445115 \dots$