Simplify the Radical Expression (4 square root of 7+3)(6 square root of 7-1)

$(4 \sqrt{7} + 3) (6 \sqrt{7} - 1)$

Step 1

Expand $(4 \sqrt{7} + 3) (6 \sqrt{7} - 1)$ using the FOIL Method.

Apply the distributive property.

$4 \sqrt{7} (6 \sqrt{7} - 1) + 3 (6 \sqrt{7} - 1)$

Apply the distributive property.

$4 \sqrt{7} (6 \sqrt{7}) + 4 \sqrt{7} \cdot - 1 + 3 (6 \sqrt{7} - 1)$

Apply the distributive property.

$4 \sqrt{7} (6 \sqrt{7}) + 4 \sqrt{7} \cdot - 1 + 3 (6 \sqrt{7}) + 3 \cdot - 1$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $4 \sqrt{7} (6 \sqrt{7})$ .

Multiply $6$ by $4$ .

$24 \sqrt{7} \sqrt{7} + 4 \sqrt{7} \cdot - 1 + 3 (6 \sqrt{7}) + 3 \cdot - 1$

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

$24 ({\sqrt{7}}^{1} \sqrt{7}) + 4 \sqrt{7} \cdot - 1 + 3 (6 \sqrt{7}) + 3 \cdot - 1$

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

$24 ({\sqrt{7}}^{1} {\sqrt{7}}^{1}) + 4 \sqrt{7} \cdot - 1 + 3 (6 \sqrt{7}) + 3 \cdot - 1$

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

$24 {\sqrt{7}}^{1 + 1} + 4 \sqrt{7} \cdot - 1 + 3 (6 \sqrt{7}) + 3 \cdot - 1$

Add $1$ and $1$ .

$24 {\sqrt{7}}^{2} + 4 \sqrt{7} \cdot - 1 + 3 (6 \sqrt{7}) + 3 \cdot - 1$

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

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

$24 {(7^{\frac{1}{2}})}^{2} + 4 \sqrt{7} \cdot - 1 + 3 (6 \sqrt{7}) + 3 \cdot - 1$

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

$24 \cdot 7^{\frac{1}{2} \cdot 2} + 4 \sqrt{7} \cdot - 1 + 3 (6 \sqrt{7}) + 3 \cdot - 1$

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

$24 \cdot 7^{\frac{2}{2}} + 4 \sqrt{7} \cdot - 1 + 3 (6 \sqrt{7}) + 3 \cdot - 1$

Cancel the common factor of $2$ .

Cancel the common factor.

$24 \cdot 7^{\frac{2}{2}} + 4 \sqrt{7} \cdot - 1 + 3 (6 \sqrt{7}) + 3 \cdot - 1$

Rewrite the expression.

$24 \cdot 7^{1} + 4 \sqrt{7} \cdot - 1 + 3 (6 \sqrt{7}) + 3 \cdot - 1$

Evaluate the exponent.

$24 \cdot 7 + 4 \sqrt{7} \cdot - 1 + 3 (6 \sqrt{7}) + 3 \cdot - 1$

Multiply $24$ by $7$ .

$168 + 4 \sqrt{7} \cdot - 1 + 3 (6 \sqrt{7}) + 3 \cdot - 1$

Multiply $- 1$ by $4$ .

$168 - 4 \sqrt{7} + 3 (6 \sqrt{7}) + 3 \cdot - 1$

Multiply $6$ by $3$ .

$168 - 4 \sqrt{7} + 18 \sqrt{7} + 3 \cdot - 1$

Multiply $3$ by $- 1$ .

$168 - 4 \sqrt{7} + 18 \sqrt{7} - 3$

Subtract $3$ from $168$ .

$165 - 4 \sqrt{7} + 18 \sqrt{7}$

Add $- 4 \sqrt{7}$ and $18 \sqrt{7}$ .

$165 + 14 \sqrt{7}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$165 + 14 \sqrt{7}$

Decimal Form:

$202.04051835 \dots$

Simplify the Radical Expression (4 square root of 7+3)(6 square root of 7-1)

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