Problem 88
Question
SIMPLIFYING EXPRESSIONS Simplify. Write your answer as a power or as an expression containing powers. (Review 8.1 ) $$4^{5} \cdot 4^{8}$$
Step-by-Step Solution
Verified Answer
The simplified form of the expression \(4^{5} \cdot 4^{8}\) is \(4^{13}\).
1Step 1: Identify the Base and the Exponents
In this problem, the base for both parts of the expression is 4, and the exponents are 5 and 8. So, we are dealing with \(4^{5} \cdot 4^{8}\).
2Step 2: Apply the Law of Exponents
According to the law of exponents, when you multiply two numbers with the same base, you can simply add the exponents. So, \(4^{5} \cdot 4^{8} = 4^{5 + 8}\).
3Step 3: Simplify the Exponents
Finally, simplify by adding the exponents 5 and 8 which gives 13. Thus, \(4^{5} \cdot 4^{8} = 4^{13}\).
Key Concepts
Law of ExponentsExponential NotationAlgebraic Expressions
Law of Exponents
Understanding the law of exponents is essential for efficiently simplifying algebraic expressions involving powers. In essence, this law provides a set of rules for performing operations on exponential notation.
One fundamental aspect of the law of exponents is dealing with multiplication when the bases are the same. When multiplying like bases, rather than multiplying the bases themselves, you simply add the exponents. This is illustrated in the exercise where, to multiply \(4^{5}\) by \(4^{8}\), you keep the base of 4 constant and add the exponents to get \(4^{5+8}\).
This rule makes calculations much simpler and quicker, especially with larger numbers or more complex expressions. Here's a quick rundown of how to apply this law:
One fundamental aspect of the law of exponents is dealing with multiplication when the bases are the same. When multiplying like bases, rather than multiplying the bases themselves, you simply add the exponents. This is illustrated in the exercise where, to multiply \(4^{5}\) by \(4^{8}\), you keep the base of 4 constant and add the exponents to get \(4^{5+8}\).
This rule makes calculations much simpler and quicker, especially with larger numbers or more complex expressions. Here's a quick rundown of how to apply this law:
Exponential Notation
Exponential notation is a convenient way to represent repeated multiplication of the same number. It is expressed as a base raised to an exponent. For example, the expression \(4^{5}\) is read as '4 to the power of 5' and signifies that 4 is multiplied by itself 4 more times, for a total of five factors of 4.
In your exercise, both \(4^{5}\) and \(4^{8}\) are written in exponential notation. It's this shorthand that makes working with very large or very small numbers possible without writing out long strings of multiplication.
Remember that exponential notation also applies to negative exponents and fractional exponents, which represent division and roots, respectively. For example, \(4^{-1} = \frac{1}{4}\) and \(4^{1/2} = \text{the square root of } 4\).
In your exercise, both \(4^{5}\) and \(4^{8}\) are written in exponential notation. It's this shorthand that makes working with very large or very small numbers possible without writing out long strings of multiplication.
Remember that exponential notation also applies to negative exponents and fractional exponents, which represent division and roots, respectively. For example, \(4^{-1} = \frac{1}{4}\) and \(4^{1/2} = \text{the square root of } 4\).
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. A key skill in algebra is simplifying these expressions to make them more understandable or to solve equations more easily.
In our case, simplifying the expression \(4^{5} \times 4^{8}\) makes the calculation more straightforward and reveals the underlying relationship between the terms involved. By consolidating the powers of 4, we achieve a simpler expression, \(4^{13}\), without changing the value of the expression.
This process of simplification often applies other rules from algebra, such as combining like terms or factoring, and is a foundational skill for advancing in mathematics. Simplifying helps not only in solving problems but also in understanding the structure and relationships within algebraic expressions.
In our case, simplifying the expression \(4^{5} \times 4^{8}\) makes the calculation more straightforward and reveals the underlying relationship between the terms involved. By consolidating the powers of 4, we achieve a simpler expression, \(4^{13}\), without changing the value of the expression.
This process of simplification often applies other rules from algebra, such as combining like terms or factoring, and is a foundational skill for advancing in mathematics. Simplifying helps not only in solving problems but also in understanding the structure and relationships within algebraic expressions.
Other exercises in this chapter
Problem 88
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