Question

Simplify each expression. $$(-1000)^{-1 / 3}$$

Short answer

-1/10

Step-by-step solution

Understand the Problem

We need to simplify the expression $$(-1000)^{-1 / 3}$$. The expression involves a negative exponent and a cube root.

Use the Negative Exponent Rule

Recall the negative exponent rule: $$a^{-n} = \frac{1}{a^n}$$. Apply this rule to the given expression: $$(-1000)^{-1/3} = \frac{1}{(-1000)^{1/3}}$$.

Simplify the Cube Root

Next, find the cube root of $$-1000$$. The cube root of $$-1000$$ is $$-10$$ because $$(-10)^3 = -10 \times -10 \times -10 = -1000$$. Therefore, $$(-1000)^{1/3} = -10$$.

Substitute and Simplify

Substitute $$(-1000)^{1/3} = -10$$ into the expression: $$\frac{1}{(-1000)^{1/3}} = \frac{1}{-10} = -\frac{1}{10}$$.

Key concepts

Negative Exponents

Negative exponents might seem a bit tricky at first, but they follow a simple rule. When you see a negative exponent like in \[a^{-n}\], it means you take the reciprocal of the base and change the exponent to positive. For instance, if you have something like \[3^{-2}\], you can convert it to \[\frac{1}{3^2}\]. This means you calculate \[3^2\] first, which is 9, then take the reciprocal, so \[\frac{1}{9}\].

The negative exponent tells us to 'flip' the base from the numerator to the denominator of a fraction, making calculations easier.

Cube Roots

Cube roots are akin to square roots but involve three equal factors. Think of the cube root of a number as answering the question: 'What number multiplied by itself three times gives us the original number?'
For example, the cube root of \[-1000\] is \[-10\]. Why? Because when you multiply -10 by itself three times, you get -1000. It’s written as \[ (-1000)^{1/3} = -10\].

Cube roots can also apply to positive numbers. For instance, the cube root of 8 is 2, because \[2 \times 2 \times 2 = 8\]. The simplicity behind cube roots helps in demystifying expressions involving roots.

Rational Exponents

Rational exponents, like \[\frac{1}{3}\], connect roots and exponents. They express roots as powers. For example, \[-1000^{\frac{1}{3}}\] is same as the cube root of -1000. Another example, \[8^{\frac{1}{3}}\], means we want the number which, when raised to the power of 3, equals 8. This is essentially saying that the cube root of 8 is 2, hence \[8^{\frac{1}{3}} = 2\].

This expression allows you to work with roots and exponents interchangeably. This makes complex calculations easier as we can leverage exponent rules to simplify our work.