Evaluate ((47^-3)^3)/((49^-22^(3/2))^2)

$\frac{{(4 \cdot 7^{- 3})}^{3}}{{(49^{- 2} \cdot 2^{\frac{3}{2}})}^{2}}$

Step 1

Simplify the numerator.

Apply the product rule to $4 \cdot 7^{- 3}$ .

$\frac{4^{3} \cdot {(7^{- 3})}^{3}}{{(49^{- 2} \cdot 2^{\frac{3}{2}})}^{2}}$

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

$\frac{64 \cdot {(7^{- 3})}^{3}}{{(49^{- 2} \cdot 2^{\frac{3}{2}})}^{2}}$

Multiply the exponents in ${(7^{- 3})}^{3}$ .

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

$\frac{64 \cdot 7^{- 3 \cdot 3}}{{(49^{- 2} \cdot 2^{\frac{3}{2}})}^{2}}$

Multiply $- 3$ by $3$ .

$\frac{64 \cdot 7^{- 9}}{{(49^{- 2} \cdot 2^{\frac{3}{2}})}^{2}}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{64 \cdot \frac{1}{7^{9}}}{{(49^{- 2} \cdot 2^{\frac{3}{2}})}^{2}}$

Raise $7$ to the power of $9$ .

$\frac{64 \cdot \frac{1}{40353607}}{{(49^{- 2} \cdot 2^{\frac{3}{2}})}^{2}}$

Step 2

Simplify the denominator.

Apply the product rule to $49^{- 2} \cdot 2^{\frac{3}{2}}$ .

$\frac{64 \cdot \frac{1}{40353607}}{{(49^{- 2})}^{2} \cdot {(2^{\frac{3}{2}})}^{2}}$

Multiply the exponents in ${(49^{- 2})}^{2}$ .

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

$\frac{64 \cdot \frac{1}{40353607}}{49^{- 2 \cdot 2} \cdot {(2^{\frac{3}{2}})}^{2}}$

Multiply $- 2$ by $2$ .

$\frac{64 \cdot \frac{1}{40353607}}{49^{- 4} \cdot {(2^{\frac{3}{2}})}^{2}}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{64 \cdot \frac{1}{40353607}}{\frac{1}{49^{4}} \cdot {(2^{\frac{3}{2}})}^{2}}$

Raise $49$ to the power of $4$ .

$\frac{64 \cdot \frac{1}{40353607}}{\frac{1}{5764801} \cdot {(2^{\frac{3}{2}})}^{2}}$

Multiply the exponents in ${(2^{\frac{3}{2}})}^{2}$ .

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

$\frac{64 \cdot \frac{1}{40353607}}{\frac{1}{5764801} \cdot 2^{\frac{3}{2} \cdot 2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{64 \cdot \frac{1}{40353607}}{\frac{1}{5764801} \cdot 2^{\frac{3}{2} \cdot 2}}$

Rewrite the expression.

$\frac{64 \cdot \frac{1}{40353607}}{\frac{1}{5764801} \cdot 2^{3}}$

Raise $2$ to the power of $3$ .

$\frac{64 \cdot \frac{1}{40353607}}{\frac{1}{5764801} \cdot 8}$

Step 3

Combine $64$ and $\frac{1}{40353607}$ .

$\frac{\frac{64}{40353607}}{\frac{1}{5764801} \cdot 8}$

Step 4

Combine $\frac{1}{5764801}$ and $8$ .

$\frac{\frac{64}{40353607}}{\frac{8}{5764801}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{64}{40353607} \cdot \frac{5764801}{8}$

Step 6

Cancel the common factor of $8$ .

Factor $8$ out of $64$ .

$\frac{8 (8)}{40353607} \cdot \frac{5764801}{8}$

Cancel the common factor.

$\frac{8 \cdot 8}{40353607} \cdot \frac{5764801}{8}$

Rewrite the expression.

$\frac{8}{40353607} \cdot 5764801$

Step 7

Cancel the common factor of $5764801$ .

Factor $5764801$ out of $40353607$ .

$\frac{8}{5764801 (7)} \cdot 5764801$

Cancel the common factor.

$\frac{8}{5764801 \cdot 7} \cdot 5764801$

Rewrite the expression.

$\frac{8}{7}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{8}{7}$

Decimal Form:

$1.\overline{142857}$

Mixed Number Form:

$1 \frac{1}{7}$

Evaluate ((4*7^-3)^3)/((49^-2*2^(3/2))^2)

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Evaluate ((47^-3)^3)/((49^-22^(3/2))^2)