Problem 9 - A Graphical Approach to College Algebra

Question

Evaluate each expression. Do not use a calculator. $$(-1000)^{2 / 3}$$

Step-by-Step Solution

Answer

\((-1000)^{2/3} = 100\).

Step 1: Understand the Expression

We want to evaluate the expression \((-1000)^{2/3}\) without using a calculator. This involves finding the cube root of \(-1000\) and then squaring the result.

Step 2: Simplify Using Exponent Law

The expression \((-1000)^{2/3}\) can be rewritten using the power of a power law: \((a^m)^n = a^{m\cdot n}\). So, \((-1000)^{2/3} = \left((-1000)^{1/3}\right)^2\).

Step 3: Evaluate the Cube Root

Find the cube root of \(-1000\), denoted as \((-1000)^{1/3}\). Since the cube root of a negative number is negative, and the cube root of \(1000\) is \(10\), we have \((-1000)^{1/3} = -10\).

Step 4: Square the Cube Root Result

Now that we have the cube root \((-10)\), square this result to evaluate \((-10)^2\). Squaring \(-10\) gives \((-10) \times (-10) = 100\).

Step 5: Conclusion

Thus, the expression \((-1000)^{2/3}\) simplifies to \(100\).

Key Concepts

Cube Root

A cube root is a special number that, when multiplied by itself twice, gives an original number. For example, the cube root of 8 is 2 because when you multiply 2 by itself three times, you get 8.
In mathematical terms, finding the cube root is the same as raising a number to the power of one-third. In symbols:

• The cube root of any number \(a\) is written as \(a^{1/3}\).

A unique property of cube roots is their behavior with negative numbers. While square roots of negative numbers are not real (they involve imaginary numbers), cube roots of negative numbers are indeed real.
For instance, the cube root of

• -1000 is -10

since

• -10 multiplied by itself twice gives -1000.

This makes cube roots flexible and useful in many mathematical scenarios.

Negative Numbers

Negative numbers are numbers less than zero and are indicated by a minus sign (-). Understanding how these numbers behave in different mathematical scenarios is crucial.
When it comes to arithmetic, there are a few key points about negative numbers to keep in mind:

• Multiplying or dividing two negative numbers results in a positive number. For example, \((-2) \times (-3) = 6\).


• Multiplying or dividing numbers with opposite signs results in a negative number, like \((-2) \times 3 = -6\).

Specifically, when calculating expressions with powers and roots, the sign of the number matters. When you take the cube root of a negative number, like with \((-1000)^{1/3}\), the result is negative because cubed negative numbers return the original negative value.
Squaring a negative number always produces a positive result, which is why

• \((-10)^2\)

results in 100. This is an important characteristic when working with exponents.

Exponent Laws

Exponent laws are rules that make complex mathematical operations simpler by providing a straightforward way to handle calculations involving powers and roots. There are several laws of exponents, but some of the most fundamental ones include:

• Power of a Power: \((a^m)^n = a^{m\cdot n}\)
• Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
• Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)

In the context of the expression \((-1000)^{2/3}\), the power of a power law allows us to rewrite the expression for easier calculation.
By expressing it as \(\left((-1000)^{1/3}\right)^2\), we can first focus on finding the cube root and then squaring the result.
Exponent laws help break down complicated expressions into more manageable parts, making them a powerful tool in mathematics.

Simplifying Expressions

Simplifying expressions involves rewriting them in a form that is easier to understand or work with. The goal is to remove unwanted parts, like overly complex fractions or redundant terms, and make calculations straightforward.
There are certain strategies often used for simplification:

• Apply the exponent laws to combine like terms or redistribute powers.
• Use arithmetic operations to reduce numbers to their simplest form.
• Combine constants and similar variable terms.

In the given problem, simplifying the expression \((-1000)^{2/3}\) meant using the laws of exponents to break it down into \(\left((-1000)^{1/3}\right)^2\).
We then separately evaluated the cube root and the square, resulting in a straightforward expression, 100, without using a calculator.
Simplification is a crucial skill in all areas of math, as it leads to quicker, accurate solutions.