Simplify the expression 15^(-2/12).

Understand the Problem

The question presents a mathematical expression involving an exponent. To understand it fully, it needs to be clarified whether the exponent is a fraction or a negative number. Assuming it is interpreted as (15^{-\frac{2}{12}}), this involves understanding negative exponents and fractional exponents (roots). The goal is to simplify this expression.

Answer

$\frac{\sqrt[6]{15^5}}{15}$
Answer for screen readers

$\frac{\sqrt[6]{15^5}}{15}$

Steps to Solve

  1. Simplify the fraction in the exponent

Simplify the fraction $-\frac{2}{12}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

$$ -\frac{2}{12} = -\frac{2 \div 2}{12 \div 2} = -\frac{1}{6} $$

  1. Rewrite the expression with the simplified exponent

Now substitute the simplified fraction back into the original expression:

$$ 15^{-\frac{2}{12}} = 15^{-\frac{1}{6}} $$

  1. Deal with the negative exponent

A negative exponent indicates that we should take the reciprocal of the base raised to the positive exponent.

$$ 15^{-\frac{1}{6}} = \frac{1}{15^{\frac{1}{6}}} $$

  1. Convert the fractional exponent to a radical

A fractional exponent of the form $\frac{1}{n}$ represents the $n$-th root. In this case, $\frac{1}{6}$ represents the 6th root.

$$ \frac{1}{15^{\frac{1}{6}}} = \frac{1}{\sqrt[6]{15}} $$

  1. Rationalize the denominator (optional)

While the expression $\frac{1}{\sqrt[6]{15}}$ is simplified, we can rationalize the denominator. To do this, we want to get rid of the radical in the denominator. We can achieve this by multiplying both the numerator and the denominator by $\sqrt[6]{15^5}$. $$ \frac{1}{\sqrt[6]{15}} \cdot \frac{\sqrt[6]{15^5}}{\sqrt[6]{15^5}} = \frac{\sqrt[6]{15^5}}{\sqrt[6]{15^6}} = \frac{\sqrt[6]{15^5}}{15} $$

$\frac{\sqrt[6]{15^5}}{15}$

More Information

The simplified form of $15^{-\frac{2}{12}}$ is $\frac{\sqrt[6]{15^5}}{15}$. This involves understanding fractional exponents as roots and negative exponents as reciprocals.

Tips

A common mistake is not simplifying the fraction in the exponent first. This can make the problem seem more complex than it is. Another mistake is misunderstanding the meaning of a negative or fractional exponent. Remember that a negative exponent implies taking the reciprocal and a fractional exponent represents a root.

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