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Chapter 5: Problem 133

Simplify each expression. (a) \((3 p)^{-2}\) (b) \(3 p^{-2}\) (c) \(-3 p^{-2}\)

Short Answer

Expert verified
(a) \frac{1}{9p^2} \), (b) \frac{3}{p^2} \), (c) \frac{-3}{p^2} \.

Step by step solution

01

Simplify (3p)^{-2}

To simplify \( (3p)^{-2} \), we use the property of exponents that \( (a \cdot b)^{-n} = a^{-n} \cdot b^{-n} \). Therefore, \((3p)^{-2} = 3^{-2} \cdot p^{-2}\). Then simplify each term: \( 3^{-2} = \frac{1}{3^2} = \frac{1}{9}\), so \((3p)^{-2} = \frac{1}{9p^2}\).
02

Simplify 3p^{-2}

For \( 3p^{-2} \), apply the negative exponent rule to \( p^{-2} \), which becomes \( \frac{1}{p^2} \). So the expression simplifies to \( 3 \cdot \frac{1}{p^2} = \frac{3}{p^2} \).
03

Simplify -3p^{-2}

Similar to step 2, simplify \( -3p^{-2} \) by applying the negative exponent rule to \( p^{-2} \), which becomes \( \frac{1}{p^2} \). Therefore, the expression simplifies to \( -3 \cdot \frac{1}{p^2} = \frac{-3}{p^2} \).

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Key Concepts

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

Negative Exponents
Understanding negative exponents is crucial in simplifying expressions. When we have a negative exponent, it means that we take the reciprocal of the base and change the exponent to positive. For example, if we have a^{-n}, it can be rewritten as \( \frac{1}{a^n} \). This rule helps us convert a seemingly complex negative exponent into a positive, more manageable one. Let's look at an example from the exercise: In (3p)^{-2}, apply the rule to both 3 and p. Rewrite this as \( \frac{1}{3^2} \cdot \frac{1}{p^2} \), which simplifies to: \( \frac{1}{9p^2} \). Breaking down negative exponents in this way simplifies working with polynomials and other algebraic expressions.
Simplifying Expressions
Simplifying expressions involves reducing them to their simplest form. This often requires using exponent rules like combining like terms, applying the distributive property, and simplifying negative exponents. Let's take another example from the exercise: 3p^{-2}. Here, p^{-2} is a negative exponent. Apply the negative exponent rule: \( p^{-2} = \frac{1}{p^2} \). So, 3p^{-2} becomes 3 \cdot \frac{1}{p^2}, which simplifies to: \( \frac{3}{p^2} \). By recognizing and properly applying these rules, you make complex expressions more straightforward and easier to manage.
Properties of Exponents
The properties of exponents are foundational in algebra. They allow us to manipulate and simplify expressions involving powers effectively. Key properties include:
  • Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
  • Power of a Power: \( (a^m)^n = a^{m\cdot n} \)
  • Power of a Product: \( (ab)^n = a^n \cdot b^n \)
  • Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
  • Negative Exponent: \( a^{-n} = \frac{1}{a^n} \)
  • Zero Exponent: \( a^0 = 1 \)
These properties make it possible to handle and simplify even more complex algebraic equations. For instance, in the exercise, (-3p^{-2}), we use the negative exponent property on p^{-2}, transforming it to \( \frac{1}{p^2} \). This leads to: -3 \cdot \frac{1}{p^2}, which simplifies to: \( \frac{-3}{p^2} \). Understanding and applying these properties can significantly enhance your ability to solve algebra problems efficiently.

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Most popular questions from this chapter

Find each product. $$ (p-3)(p+3) $$

Multiply each pair of conjugates using the Product of Conjugates Pattern. $$ \left(15 m^{2}-8 n^{4}\right)\left(15 m^{2}+8 n^{4}\right) $$

Find each product. $$ 5 q^{3}\left(q^{2}-2 q+6\right) $$

Square each binomial using the Binomial Squares Pattern. $$ \left(\frac{1}{5} x-\frac{1}{7} y\right)^{2} $$

Multiply the binomials. Use any method. $$ (3 r s-7)(3 r s-4) $$
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