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Chapter 10: Problem 35

Rewrite with a positive exponent and evaluate. \(1000^{-1 / 3}\)

Short Answer

Expert verified
The given expression \(1000^{-1/3}\) can be rewritten with a positive exponent as \(\frac{1}{1000^{1/3}}\), which simplifies to \(\frac{1}{\sqrt[3]{1000}}\). The cube root of 1000 is 10, so the final answer is \(\frac{1}{10}\).

Step by step solution

01

Rewrite the negative exponent as a positive exponent

To rewrite the given expression \(1000^{-1/3}\) with a positive exponent, use the property \(a^{-n} = \frac{1}{a^n}\). So, we have: \(1000^{-1/3} = \frac{1}{1000^{1/3}}\)
02

Rewrite the exponent as a radical

We can rewrite the expression as a cube root to simplify further: \(\frac{1}{1000^{1/3}} = \frac{1}{\sqrt[3]{1000}}\)
03

Simplify the cube root

Now, we need to find the cube root of 1000: \(\sqrt[3]{1000} = 10\) since \(10^3 = 1000\).
04

Write the final answer

Put the value of the cube root back into the expression: \(\frac{1}{\sqrt[3]{1000}} = \frac{1}{10}\) So, \(1000^{-1/3} = \frac{1}{10}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Negative Exponents
A negative exponent might seem a bit confusing at first, but it's straightforward once you get the hang of it. When you see a negative exponent, it's telling you to take the reciprocal of the base raised to the opposite exponent. For example, if you have a number like \(a^{-n}\), this is equivalent to \(\frac{1}{a^n}\). So instead of viewing it as a negative, you flip the base to its reciprocal with a positive exponent.

This concept helps in simplifying expressions and working without large numbers as exponents become easily manageable. Let's take \(1000^{-1/3}\) as an example. It is the same as \(\frac{1}{1000^{1/3}}\), which makes calculations and understanding more intuitive.
Cube Roots
Cube roots are an intriguing part of radical expressions concerning finding a number that, when multiplied by itself three times, gives the original number. The cube root of a number \(n\) is expressed as \(\sqrt[3]{n}\).

In our example, to find the cube root of 1000 means looking for a number that, when cubed, equals 1000. The answer is 10, as \(10^3 = 1000\). Understanding cube roots is essential in solving various mathematical problems as they allow you to simplify expressions involving cubic powers.
  • Cube roots are related to geometry and volume calculations.
  • Useful in transforming large numbers into manageable figures.
  • Crucial in real-world applications like physics and engineering.
Radical Expressions
Radical expressions involve roots and can sometimes be daunting to simplify, especially when they involve higher order roots like cube roots. A radical is an expression that includes a root, and when you're working with them, it usually involves solving or simplifying them to a more digestible form.

The expression \(\sqrt[3]{1000}\) allows you to comprehend and work with the number without calculating powers beyond your current capabilities. Radical expressions find applications beyond mathematics, notably in physics and chemistry where quantities often need to be expressed in terms of roots.
  • They represent irrational numbers in a simplified format.
  • Often simplify complex calculations.
  • Used widely in calculus and algebra.
Simplifying Radicals
Simplifying radicals is crucial for making complex radicands (the number inside the radical) more approachable. By breaking down a root into an easier-to-evaluate form, you reduce the problem size and enhance understanding.

In our example, simplifying \(\sqrt[3]{1000}\) yielded 10, which made calculating the overall expression much simpler. Breaking down numbers into prime factors and using them to simplify expressions often help in tackling radical expressions.
  • This technique is indispensable in algebra for solving equations.
  • Simplification leads to more accurate calculations.
  • Simplifying radicals often serves as a foundational step in more advanced math problems.

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