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Chapter 2: Problem 95

Simplify each of the following. $$5^{2}$$

Short Answer

Expert verified
The simplified form of \(5^2\) is 25.

Step by step solution

01

Understand the Expression

The expression \(5^2\) means 5 raised to the power of 2. This implies we need to multiply 5 by itself.
02

Perform the Multiplication

Multiply 5 by itself: \(5 \times 5 = 25\).
03

Write the Simplified Result

The simplified form of \(5^2\) is 25.

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Key Concepts

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

Understanding Powers
In mathematics, powers are a way to express repeated multiplication of the same number. We write a number raised to a power using the format \( a^b \), where \( a \) is known as the base, and \( b \) is the exponent or power. This tells us how many times the base number is multiplied by itself.
For instance, in the expression \( 5^2 \), 5 is the base, and 2 is the exponent. Here, the exponent 2 indicates that the number 5 is multiplied by itself once:
  • For any base \( a \) raised to the first power, \( a^1 \), the result is the number itself.
  • Raising a number to the power of zero, \( a^0 \), equals 1, assuming \( a eq 0 \).
  • Powers allow us to express calculations of large numbers succinctly.
The Process of Multiplication Involved in Powers
When dealing with powers, multiplication is at the core of solving these expressions. To compute \( a^b \), you multiply the base \( a \) by itself \( b \) times. This reinforces the idea of powers as a form of repeated multiplication.
  • In the case of \( 5^2 \), you multiply 5 by 5, giving the result \( 5 \times 5 = 25 \).
  • For larger powers, such as \( 5^4 \), you would multiply 5 by itself three more times: \( 5 \times 5 \times 5 \times 5 \).
  • Each multiplication corresponds to one instance of multiplying the base by itself.
Understanding this process of multiplication is crucial for further simplifying powers.
Simplification of Powers
Simplification in mathematics means breaking down complex expressions into their simplest form. When you simplify a power, such as \( 5^2 \), you're looking for the most basic way to express the product of repeated multiplication.
  • Once the multiplication is performed, the result is a single number. For \( 5^2 \), the simplified form is 25.
  • Simplification helps make expressions easier to use in equations or comparisons by reducing them to basic arithmetic.
  • It provides clarity, allowing one to easily understand the magnitude or size of the expression compared to others.
By simplifying, we transform the exponential expression into the more accessible product form, useful for further mathematical operations.

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