Simplify ( square root of 343)/( cube root of 49)

$\frac{\sqrt{343}}{\sqrt[3]{49}}$

Step 1

Simplify the numerator.

Rewrite $343$ as $7^{2} \cdot 7$ .

Factor $49$ out of $343$ .

$\frac{\sqrt{49 (7)}}{\sqrt[3]{49}}$

Rewrite $49$ as $7^{2}$ .

$\frac{\sqrt{7^{2} \cdot 7}}{\sqrt[3]{49}}$

Pull terms out from under the radical.

$\frac{7 \sqrt{7}}{\sqrt[3]{49}}$

Step 2

Multiply $\frac{7 \sqrt{7}}{\sqrt[3]{49}}$ by $\frac{{\sqrt[3]{49}}^{2}}{{\sqrt[3]{49}}^{2}}$ .

$\frac{7 \sqrt{7}}{\sqrt[3]{49}} \cdot \frac{{\sqrt[3]{49}}^{2}}{{\sqrt[3]{49}}^{2}}$

Step 3

Simplify terms.

Combine and simplify the denominator.

Multiply $\frac{7 \sqrt{7}}{\sqrt[3]{49}}$ by $\frac{{\sqrt[3]{49}}^{2}}{{\sqrt[3]{49}}^{2}}$ .

$\frac{7 \sqrt{7} {\sqrt[3]{49}}^{2}}{\sqrt[3]{49} {\sqrt[3]{49}}^{2}}$

Raise $\sqrt[3]{49}$ to the power of $1$ .

$\frac{7 \sqrt{7} {\sqrt[3]{49}}^{2}}{{\sqrt[3]{49}}^{1} {\sqrt[3]{49}}^{2}}$

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

$\frac{7 \sqrt{7} {\sqrt[3]{49}}^{2}}{{\sqrt[3]{49}}^{1 + 2}}$

Add $1$ and $2$ .

$\frac{7 \sqrt{7} {\sqrt[3]{49}}^{2}}{{\sqrt[3]{49}}^{3}}$

Rewrite ${\sqrt[3]{49}}^{3}$ as $49$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{49}$ as $49^{\frac{1}{3}}$ .

$\frac{7 \sqrt{7} {\sqrt[3]{49}}^{2}}{{(49^{\frac{1}{3}})}^{3}}$

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

$\frac{7 \sqrt{7} {\sqrt[3]{49}}^{2}}{49^{\frac{1}{3} \cdot 3}}$

Combine $\frac{1}{3}$ and $3$ .

$\frac{7 \sqrt{7} {\sqrt[3]{49}}^{2}}{49^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{7 \sqrt{7} {\sqrt[3]{49}}^{2}}{49^{\frac{3}{3}}}$

Rewrite the expression.

$\frac{7 \sqrt{7} {\sqrt[3]{49}}^{2}}{49^{1}}$

Evaluate the exponent.

$\frac{7 \sqrt{7} {\sqrt[3]{49}}^{2}}{49}$

Cancel the common factor of $7$ and $49$ .

Factor $7$ out of $7 \sqrt{7} {\sqrt[3]{49}}^{2}$ .

$\frac{7 (\sqrt{7} {\sqrt[3]{49}}^{2})}{49}$

Cancel the common factors.

Factor $7$ out of $49$ .

$\frac{7 (\sqrt{7} {\sqrt[3]{49}}^{2})}{7 \cdot 7}$

Cancel the common factor.

$\frac{7 (\sqrt{7} {\sqrt[3]{49}}^{2})}{7 \cdot 7}$

Rewrite the expression.

$\frac{\sqrt{7} {\sqrt[3]{49}}^{2}}{7}$

Step 4

Simplify the numerator.

Rewrite ${\sqrt[3]{49}}^{2}$ as ${(49^{2})}^{\frac{1}{3}}$ .

$\frac{\sqrt{7} \sqrt[3]{49^{2}}}{7}$

Raise $49$ to the power of $2$ .

$\frac{\sqrt{7} \sqrt[3]{2401}}{7}$

Rewrite $2401$ as $7^{3} \cdot 7$ .

Factor $343$ out of $2401$ .

$\frac{\sqrt{7} \sqrt[3]{343 (7)}}{7}$

Rewrite $343$ as $7^{3}$ .

$\frac{\sqrt{7} \sqrt[3]{7^{3} \cdot 7}}{7}$

Pull terms out from under the radical.

$\frac{\sqrt{7} \cdot 7 \sqrt[3]{7}}{7}$

Combine exponents.

Rewrite the expression using the least common index of $6$ .

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

$\frac{7 (7^{\frac{1}{2}} \sqrt[3]{7})}{7}$

Rewrite $7^{\frac{1}{2}}$ as $7^{\frac{3}{6}}$ .

$\frac{7 (7^{\frac{3}{6}} \sqrt[3]{7})}{7}$

Rewrite $7^{\frac{3}{6}}$ as $\sqrt[6]{7^{3}}$ .

$\frac{7 (\sqrt[6]{7^{3}} \sqrt[3]{7})}{7}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$\frac{7 (\sqrt[6]{7^{3}} \cdot 7^{\frac{1}{3}})}{7}$

Rewrite $7^{\frac{1}{3}}$ as $7^{\frac{2}{6}}$ .

$\frac{7 (\sqrt[6]{7^{3}} \cdot 7^{\frac{2}{6}})}{7}$

Rewrite $7^{\frac{2}{6}}$ as $\sqrt[6]{7^{2}}$ .

$\frac{7 (\sqrt[6]{7^{3}} \sqrt[6]{7^{2}})}{7}$

Combine using the product rule for radicals.

$\frac{7 \sqrt[6]{7^{3} \cdot 7^{2}}}{7}$

Multiply $7^{3}$ by $7^{2}$ by adding the exponents.

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

$\frac{7 \sqrt[6]{7^{3 + 2}}}{7}$

Add $3$ and $2$ .

$\frac{7 \sqrt[6]{7^{5}}}{7}$

Raise $7$ to the power of $5$ .

$\frac{7 \sqrt[6]{16807}}{7}$

Step 5

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 \sqrt[6]{16807}}{7}$

Divide $\sqrt[6]{16807}$ by $1$ .

$\sqrt[6]{16807}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt[6]{16807}$

Decimal Form:

$5.06114018 \dots$

Simplify ( square root of 343)/( cube root of 49)

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