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Chapter 2: Problem 41

Simplify the given expression as completely as possible. $$a^{2} a^{7}$$

Short Answer

Expert verified
The simplified expression is \(a^{9}\).

Step by step solution

01

Identify the expression components

The expression given is a product of two powers of the same base, which are \(a^{2}\) and \(a^{7}\).
02

Apply the product of powers property

Use the property \(a^{m} \times a^{n} = a^{m+n}\) to simplify the expression. In this case, the base \(a\) will remain the same, and the exponents will be added.
03

Perform the addition of exponents

Add the exponents: \(2 + 7 = 9\). So, \(a^{2} a^{7} = a^{9}\).

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Key Concepts

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

Exponents
Exponents are a way to represent repeated multiplication of the same number. For example, \(a^3\) means \(a \times a \times a\). Here, the number \(a\) is called the 'base,' and the number 3 is called the 'exponent.' Exponents make it easier to write and work with very large or very small numbers.

When working with exponents, there are several key rules to remember:
  • \(a^m \times a^n = a^{m+n}\) (Product of Powers)
  • \(\frac{a^m}{a^n} = a^{m-n}\) (Quotient of Powers)
  • \((a^m)^n = a^{m \times n}\) (Power of a Power)
These rules simplify the process of working with expressions that contain exponents. Understanding these rules is essential for solving algebraic expressions.
Simplifying Expressions
Simplifying an algebraic expression means rewriting it in the simplest form possible. This involves combining like terms and using algebraic rules effectively.

For an expression with exponents, like \(a^2 a^7\), we aim to combine the terms into one simple term. This makes the expression easier to read and work with in future calculations.

Here are steps to simplify expressions with exponents:
  • Identify the terms with the same base.
  • Use the appropriate properties of exponents.
  • Combine the exponents as needed.
For example, simplifying \(a^2 a^7\) involves using the product of powers property to add the exponents, resulting in \(a^9\).
Product of Powers Property
The product of powers property is a fundamental rule in algebra. It states that when you multiply two powers that have the same base, you add the exponents.

Mathematically, \(a^m \times a^n = a^{m+n}\), where \(a\) is the base and \(m\) and \(n\) are the exponents.

Let's break down the problem:
  • The given expression is \(a^2 a^7\).
  • Here, the base is \(a\).
  • We apply the product of powers property: \(2 + 7 = 9\).
  • The simplified form is \(a^9\).
This property is very useful when dealing with larger algebraic expressions involving exponents. By mastering this, you can simplify and solve many complex equations more efficiently.

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Most popular questions from this chapter

Explain what is wrong with each of the following: (a) \(3 \cdot 2^{2} \stackrel{?}{=} 6^{2} \stackrel{?}{=} 36\) (b) \(3 x^{4} \stackrel{?}{=} 3 x \cdot 3 x \cdot 3 x \cdot 3 x\) (c) \(5^{4} \cdot 2 \stackrel{?}{=} 20 \cdot 2 \stackrel{?}{=}40\) (d) \(\left(3 x^{3}\right)^{2} \stackrel{?}{=} 6 x^{6}\) (e) \(-3^{4} \stackrel{?}{=}(-3) \cdot(-3) \cdot(-3)(-3) \stackrel{?}{=} 81\) (f) \(x^{4}+x^{5} \stackrel{?}{=} x^{9}\) (g) \(x^{2} \cdot x^{7} \stackrel{?}{=} x^{14}\) (h) \(\left(x^{2}\right)^{4} \stackrel{?}{=} x^{6}\)

Simplify each expression as completely as possible. $$8 z^{2}\left[z^{2}-z(z-6)\right]-7 z\left(z^{2}-3 z\right)$$

Simplify by combining like terms whenever possible. $$4(a+b)+4 a-b$$

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$x-(y-z)$$

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$3 x^{2}+4 x+1$$
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