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Chapter 9: Problem 16

Write each expression with positive exponents, then simplify. $$X^{-3} \cdot X^{-7}$$

Short Answer

Expert verified
\( \frac{1}{X^{10}} \)

Step by step solution

01

Apply the Rule of Exponents

When multiplying two powers with the same base, you add the exponents. Here, we have the expression \( X^{-3} \cdot X^{-7} \). According to the rule of exponents, we calculate as follows: \[ X^{-3} \cdot X^{-7} = X^{-3 + (-7)} = X^{-10} \]
02

Convert to Positive Exponents

To express the result with positive exponents, recall that \( X^{-a} = \frac{1}{X^a} \). Therefore, \[ X^{-10} = \frac{1}{X^{10}} \]
03

Simplify the Expression

The expression is already simplified once it has positive exponents. The final expression is \[ \frac{1}{X^{10}} \]

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Key Concepts

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

Understanding Negative Exponents
Negative exponents appear in mathematics and often confuse many students. But they are simpler to understand than you might think! A negative exponent indicates that the base is on the denominator of a fraction. Essentially, it means "take the reciprocal of the base raised to the positive version of the exponent." For example, when you see an expression like \( X^{-a} \), you can rewrite it as \( \frac{1}{X^a} \).
  • Imagine if you had \( 2^{-3} \). Using the rule, it's \( \frac{1}{2^3} \). That simplifies to \( \frac{1}{8} \).
  • It's important to know that negative exponents don't mean negative numbers. They simply express a different mathematical transformation called a reciprocal.
  • By converting negative exponents to positive ones, it makes expressions easier to handle and understand, like in our original exercise!
Multiplication of Powers with the Same Base
Multiplication of powers with the same base is a fundamental rule in exponents. It states when you multiply powers having the same base, you simply add the exponents together. This rule can simplify complex expressions and reduce errors.
  • The general form for this rule is \( X^a \cdot X^b = X^{a+b} \).
  • In the given exercise, we multiply \( X^{-3} \) by \( X^{-7} \). Using our rule, we add the exponents: \( -3 \) plus \( -7 \) equals \( -10 \), so \( X^{-3} \cdot X^{-7} = X^{-10} \).
  • Recognizing patterns like this helps in simplifying even larger expressions and understanding exponential models in mathematics and science.
Simplifying Expressions with Exponents
Simplifying expressions is like cleaning up a messy room. It makes them neater and easier to handle! When simplifying expressions with exponents, our goal is to make them as straightforward as possible.
  • First, apply any rules of exponents needed, like the multiplication rule in our exercise.
  • Next, convert any negative exponents to positive ones. For \( X^{-10} \), this would transform it into \( \frac{1}{X^{10}} \).
  • This final expression, \( \frac{1}{X^{10}} \), is considered simplified because it uses positive exponents and cannot be reduced further.
  • Regular practice in simplifying helps build confidence and efficiency, especially when dealing with more complex algebraic expressions.

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