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$Express the following fraction in simplest form, only using positive exponents. ((-2w^(3))^(2))/(10w^(-2)) Answer:$

Express the following fraction in simplest form, only using positive exponents. $\newline$ $\frac{\left(-2 w^{3}\right)^{2}}{10 w^{-2}}$ $\newline$ Answer:

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Math Problems

Algebra 1

Multiplication with rational exponents

Full solution

Q. Express the following fraction in simplest form, only using positive exponents.

\newline

\frac{\left(-2 w^{3}\right)^{2}}{10 w^{-2}}

\newline

Answer:

Write Expression, Identify Exponent: Write down the given expression and identify the negative exponent. $\newline$ The given expression is $((-2w^{3})^{2})/(10w^{-2})$ . $\newline$ We need to simplify this expression and make sure all exponents are positive.
Apply Power of a Power Rule: Apply the power of a power rule to the numerator. $\newline$ The power of a power rule states that $(a^m)^n = a^{m*n}$ . So we apply this rule to the numerator: $\newline$ $((-2w^{3})^{2}) = (-2)^2 \times (w^{3})^2 = 4w^{3*2} = 4w^6$ .
Convert Negative Exponent: Convert the negative exponent in the denominator to a positive exponent. $\newline$ To convert a negative exponent to a positive exponent, we take the reciprocal of the base. So for $10w^{-2}$ , we can write it as $\frac{10}{w^2}$ .
Combine Numerator and Denominator: Combine the numerator and the new denominator. $\newline$ Now we have $4w^6$ in the numerator and $\frac{10}{w^2}$ in the denominator. We can write this as: $\newline$ $(4w^6) / (\frac{10}{w^2}) = (4w^6) * (\frac{w^2}{10})$ .
Simplify by Multiplying Terms: Simplify the expression by multiplying the terms. $\newline$ Now we multiply the numerators and the denominators: $\newline$ $(4w^6) \times (\frac{w^2}{10}) = (4 \times w^{6+2}) / 10 = \frac{4w^8}{10}$ .
Simplify Fraction: Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor. $\newline$ The greatest common divisor of $4$ and $10$ is $2$ . So we divide both the numerator and the denominator by $2$ : $\newline$ $\frac{4w^8}{10} = \frac{2w^8}{5}$ .