$$ \frac { 27 ^ { - \frac { 1 } { 3 } } imes 625 ^ { \overline { 4 } } } { \sqrt[ 5 ] { 243 } imes \sqrt[ 3 ] { 125 } } = $$

Question

$$ \frac { 27 ^ { - \frac { 1 } { 3 } } 	imes 625 ^ { \overline { 4 } } } { \sqrt[ 5 ] { 243 } 	imes \sqrt[ 3 ] { 125 } } = $$

UpStudy AI · Accepted Answer

Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt[5]{243}$$
- step1: Write in exponential form:
  $$\sqrt[5]{3^{5}}$$
- step2: Simplify the root:
  $$3$$
Calculate or simplify the expression $$ \sqrt[ 3 ] { 125 } $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt[3]{125}$$
- step1: Write in exponential form:
  $$\sqrt[3]{5^{3}}$$
- step2: Simplify the root:
  $$5$$
Calculate or simplify the expression $$ 625 ^ { 4 } $$.
Calculate the value by following steps:
- step0: Calculate:
  $$625^{4}$$
Calculate or simplify the expression $$ 27 ^ { - \frac { 1 } { 3 } } $$.
Calculate the value by following steps:
- step0: Calculate:
  $$27^{-\frac{1}{3}}$$
- step1: Express with a positive exponent:
  $$\frac{1}{27^{\frac{1}{3}}}$$
- step2: Evaluate the power:
  $$\frac{1}{3}$$
Calculate or simplify the expression $$ \frac { 0.3 \times 1.525879 \times 10^{11} } { 3 \times 5 } $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.3\times 1.525879\times 10^{11}}{3\times 5}$$
- step1: Multiply the terms:
  $$\frac{0.4577637\times 10^{11}}{3\times 5}$$
- step2: Multiply the numbers:
  $$\frac{0.4577637\times 10^{11}}{15}$$
- step3: Factor the expression:
  $$\frac{0.4577637\times 5^{11}\times 2^{11}}{15}$$
- step4: Factor the expression:
  $$\frac{0.4577637\times 5^{11}\times 2^{11}}{5\times 3}$$
- step5: Reduce the fraction:
  $$\frac{0.4577637\times 5^{10}\times 2^{11}}{3}$$
- step6: Simplify:
  $$\frac{9155274000}{3}$$
- step7: Reduce the numbers:
  $$\frac{3051758000}{1}$$
- step8: Calculate:
  $$3051758000$$
Let's solve the expression step by step:

The expression we need to evaluate is:

$$
\frac { 27 ^ { - \frac { 1 } { 3 } } \times 625 ^ { 4 } } { \sqrt[ 5 ] { 243 } \times \sqrt[ 3 ] { 125 } }
$$

### Step 1: Calculate $$ 27^{-\frac{1}{3}} $$

We know that:

$$
27 = 3^3
$$

Thus,

$$
27^{-\frac{1}{3}} = (3^3)^{-\frac{1}{3}} = 3^{-1} = \frac{1}{3} \approx 0.3
$$

### Step 2: Calculate $$ 625^4 $$

We can express $$ 625 $$ as:

$$
625 = 5^4
$$

Therefore,

$$
625^4 = (5^4)^4 = 5^{16} \approx 1.525879 \times 10^{11}
$$

### Step 3: Calculate $$ \sqrt[5]{243} $$

We know that:

$$
243 = 3^5
$$

Thus,

$$
\sqrt[5]{243} = 3
$$

### Step 4: Calculate $$ \sqrt[3]{125} $$

We know that:

$$
125 = 5^3
$$

Thus,

$$
\sqrt[3]{125} = 5
$$

### Step 5: Substitute values into the expression

Now we can substitute these values back into the original expression:

$$
\frac { 0.3 \times 1.525879 \times 10^{11} } { 3 \times 5 }
$$

Calculating the denominator:

$$
3 \times 5 = 15
$$

### Step 6: Final Calculation

Now we can compute the final value:

$$
\frac { 0.3 \times 1.525879 \times 10^{11} } { 15 }
$$

Calculating this gives:

$$
\approx 3051758000 \quad \text{or} \quad 3.051758 \times 10^{9}
$$

### Final Answer

Thus, the final result is:

$$
\frac { 27 ^ { - \frac { 1 } { 3 } } \times 625 ^ { 4 } } { \sqrt[ 5 ] { 243 } \times \sqrt[ 3 ] { 125 } } \approx 3.051758 \times 10^{9}
$$
The result of the expression is approximately $$ 3.051758 \times 10^{9} $$.

\( \frac { 27 ^ { - \frac { 1 } { 3 } } \times 625 ^ { \overline { 4 } } } { \sqrt[ 5 ] { 243 } \times \sqrt[ 3 ] { 125 } } = \)

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