Problem 543 - Elementary Algebra

Question

In the following exercises, simplify. (a) \((-1000)^{\frac{1}{3}}\) (b) \(-1000^{\frac{1}{3}}\) (c) (1000) \(^{-\frac{1}{3}}\)

Step-by-Step Solution

Answer

(a) -10, (b) -10, (c) 0.1

Step 1: Simplify \((-1000)^{\frac{1}{3}}\)

To find \((-1000)^{\frac{1}{3}}\), we need to find the cube root of -1000. The cube root of -1000 is the number that, when multiplied by itself three times, equals -1000. Since \(-10 \times -10 \times -10 = -1000\), the cube root of -1000 is \-10\. Therefore, \((-1000)^{\frac{1}{3}} = -10\).

Step 2: Simplify \(-1000^{\frac{1}{3}}\)

First, find the cube root of 1000. The cube root of 1000 is the number that, when multiplied by itself three times, equals 1000. Since \(10 \times 10 \times 10 = 1000\), the cube root of 1000 is 10. Therefore, \(-1000^{\frac{1}{3}} = -10\).

Step 3: Simplify \(1000^{-\frac{1}{3}}\)

To simplify \(1000^{-\frac{1}{3}}\), recall that a negative exponent indicates the reciprocal of the base raised to the positive exponent. Hence, \(\frac{1}{1000^{\frac{1}{3}}}\). The cube root of 1000 is 10 (since \(10 \times 10 \times 10 = 1000\)), so \(\frac{1}{1000^{\frac{1}{3}}} = \frac{1}{10}\) or 0.1.

Key Concepts

Simplifying Cube Roots

A cube root is a number that, when multiplied by itself three times, gives the original number. For instance, the cube root of 8 is 2, because when you multiply 2 by itself three times, you get 8:

• 2 × 2 × 2 = 8

To find the cube root, look for a number that works in this way.
Here's how you would approach the cube root of -1000:

• Find a number that when multiplied by itself three times equals -1000.
• -10 × -10 × -10 equals -1000.

So, the cube root of -1000 is -10. We write this as: \((-1000)^{\frac{1}{3}} = -10\).
Notice the negative sign is inside the parentheses. This affects the final result and keeps the answer negative.

Understanding Negative Exponents

Negative exponents might look tricky, but they're quite simple. A negative exponent tells you to take the reciprocal of the base number and then apply the positive exponent. For example:

• \(a^{-n} = \frac{1}{a^n}\)

Let's apply this to 1000 raised to the power of \(-\frac{1}{3}\):

• 1000^{-\frac{1}{3}}ewline
• First, calculate the positive exponent result, \1000^{\frac{1}{3}}\, which is the cube root of 1000. \( \text{Since, } 10 \times 10 \times 10 = 1000\)
• Thus, 1000^{\frac{1}{3}} = 10.ewline
• Next, apply the negative exponent: \1000^{-\frac{1}{3}} = \frac{1}{1000^{\frac{1}{3}}} = \frac{1}{10}=0.1\

So, \1000^{-\frac{1}{3}} = 0.1\.

Mastering Exponent Rules

Exponent rules are fundamental in simplifying expressions involving powers. Here are some of the key rules to keep in mind:

• Product of Powers: \(a^m \times a^n = a^{m+n}\), where you add the exponents if the bases are the same.
• Power of a Power: \((a^m)^n = a^{m \times n}\), where you multiply the exponents.
• Power of a Product: \((ab)^n = a^n \times b^n\), where the exponent applies to both factors inside the parentheses.
• Power of a Quotient: \(\frac{a^m}{a^n} = a^{m-n}\), where you subtract the exponents if dividing the same base.
• Zero Exponent: \(a^0 = 1\), any number raised to the zero power is 1.
• Negative Exponent: \(a^{-n} = \frac{1}{a^n}\), as we've just discussed.

These rules help in simplifying complex expressions and solving exponent problems efficiently. Always take it step by step, and use these rules to break down problems into manageable parts.