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Chapter 7: Problem 55

Simplify each number. $$100^{4.5}$$

Short Answer

Expert verified
The simplified form of \(100^{4.5}\) is 1000000000.

Step by step solution

01

Understanding Exponential Expression

Rewrite the expression \(100^{4.5}\) as \(100^{4+0.5}\). This allows us to perform operations on \(100^4\) and \(100^{0.5}\) separately.
02

Solve for \(100^4\)

The base is 100 and the exponent is 4. So \(100^4\) equals 100*100*100*100, which equals 100000000.
03

Solve for \(100^{0.5}\)

The base is 100 and the exponent is 0.5, which is the same as taking the square root. So \(\sqrt{100}\) equals 10.
04

Combine the results

Multiply the results of the two previous steps: 100000000 times 10 equals 1000000000.

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Key Concepts

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

Exponentiation
Exponentiation is a way to express repeated multiplication of the same number. It involves two components: the base and the exponent. The base is the number you want to multiply, while the exponent tells you how many times to multiply the base by itself. For example, in the expression \(100^4\), 100 is the base, and 4 is the exponent, meaning you multiply 100 by itself four times:
  • \(100 \times 100 \times 100 \times 100 = 100000000\).
The exponent can also be a fraction, like 0.5 in \(100^{0.5}\), which relates to square roots. Understanding exponentiation requires recognizing that the base is repeatedly multiplied. This concept allows for simplifying large numbers efficiently and is a cornerstone of many math operations.
Square Roots
The square root is a special type of exponentiation where the exponent is always 0.5. It asks, "What number multiplied by itself yields the given number?" In the example \(100^{0.5}\), we are looking for a number that multiplied by itself equals 100. This number is 10 because \(10 \times 10 = 100\). The operation of taking a square root simplifies expressions where the exponent is a fraction. It is often an essential part of simplifying expressions that involve non-integer exponents. When simplifying an expression like \(100^{4.5}\), recognizing \(100^{0.5}\) as the square root of 100 can help by reducing it to its simplest form of 10.
Simplification of Numbers
Simplification of numbers involves reducing expressions to their most basic form, where they are easier to understand or work with. This often involves
  • recognizing the parts of the expression that can be simplified independently,
  • applying mathematical rules and properties like those of exponents or roots.
In our exercise, simplifying \(100^{4.5}\) requires splitting it into \(100^4\) and \(100^{0.5}\) because there are rules for handling each part that make the entire expression easier to compute:
  • Calculate \(100^4 = 100000000\).
  • Find \(100^{0.5} = 10\).
Finally, multiply these results to get simplified expression: 1000000000. Simplification reduces complex expressions to simpler or more recognizable forms, assisting in both computation and comprehension.

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