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Rewrite the expression as a product of identical factors to find the product for 100 3/2

. Submit your answer in simplified form.

11 answers

To rewrite the expression 100^(3/2) as a product of identical factors, we can break it down as follows:

  1. Start with \(100^{3/2}\): \[ 100^{3/2} = (100^1)^{3/2} \] This can be rewritten as: \[ (100^{1/2})^3 \]

  2. Calculate \(100^{1/2}\): \[ 100^{1/2} = \sqrt{100} = 10 \]

  3. Substitute back into the expression: \[ (100^{1/2})^3 = (10)^3 \]

  4. Now calculate \(10^3\): \[ 10^3 = 10 \times 10 \times 10 = 1000 \]

So, \(100^{3/2}\) can be rewritten as:

\[ (10)^3 \]

The simplified form of the product is:

\[ \boxed{1000} \]

Simplify using the denominator of the exponent before applying the numerator on the exponent.

To simplify \(100^{3/2}\) using the denominator of the exponent before applying the numerator, we can follow these steps:

  1. Rewrite the expression using the properties of exponents: \[ 100^{3/2} = (100^{1/2})^3 \]

  2. Calculate \(100^{1/2}\), which refers to the square root of 100: \[ 100^{1/2} = \sqrt{100} = 10 \]

  3. Now apply the exponent of 3: \[ (10)^3 = 10 \times 10 \times 10 = 1000 \]

Thus, another way to express the process directly through fractional exponents can be seen as:

\[ 100^{3/2} = (100^{1/2})^3 = 10^3 = 1000 \]

Therefore, in simplified form, the answer remains:

\[ \boxed{1000} \]

This method emphasizes the use of the denominator in the exponent first (the square root) before applying the numerator.

so what is the right answer

The right answer, after simplifying \(100^{3/2}\), is:

\[ \boxed{1000} \]

This represents \(100^{3/2}\) simplified fully, confirming that the value is indeed 1000.

Rewrite the expression as a product of identical factors to simplify 125 2/3
. Submit your answer in simplified form.

To rewrite the expression \(125^{2/3}\) as a product of identical factors, we can follow these steps:

  1. Rewrite the expression: \[ 125^{2/3} = (125^{1/3})^2 \]

  2. Calculate \(125^{1/3}\), which refers to the cube root of 125: \[ 125^{1/3} = \sqrt[3]{125} = 5 \] (since \(5^3 = 125\))

  3. Now apply the exponent of 2: \[ (5)^2 = 5 \times 5 = 25 \]

Thus, rewriting \(125^{2/3}\) as a product of identical factors gives us:

\[ (125^{1/3})^2 = (5)^2 = 25 \]

Therefore, in simplified form, the answer is:

\[ \boxed{25} \]

What rational number could be used as an exponent to rewrite 23−−√4
?(1 point)

To rewrite \( \sqrt[4]{23} \) using a rational exponent, we can express the fourth root as an exponent. The fourth root of a number can be represented as raising that number to the power of \( \frac{1}{4} \).

Therefore, the expression can be rewritten as:

\[ \sqrt[4]{23} = 23^{1/4} \]

This means the rational number that could be used as an exponent to rewrite \( \sqrt[4]{23} \) is:

\[ \boxed{\frac{1}{4}} \]

The denominator of the rational exponent should match the type of root. What will the numerator be?

In the expression \( \sqrt[4]{23} \) rewritten using a rational exponent, we have previously expressed it as \( 23^{1/4} \).

In terms of a rational exponent of the form \( \frac{a}{b} \), where \( b \) corresponds to the type of root (in this case, a fourth root), the denominator \( b \) is 4.

The numerator \( a \) corresponds to the power to which 23 is raised when it is expressed outside of the root. Since there is no additional power applied in this case, the exponent can be simplified as:

\[ 23^{1/4} = 23^{\frac{1}{4}} \]

Here, the numerator \( a \) is 1.

Therefore, in summary:

  • Denominator corresponds to the root type: \( 4 \) (for fourth root).
  • Numerator: \( 1 \).

So the answer for the numerator is:

\[ \boxed{1} \]