Problem 32 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \(\sqrt{100}+\sqrt{81}-4^{2}\)

Step-by-Step Solution

Answer

The value is 3.

Step 1: Calculate the Square Root of 100

The square root of 100 is calculated by finding a number that, when multiplied by itself, gives 100. Therefore, \( \sqrt{100} = 10 \).

Step 2: Calculate the Square Root of 81

The square root of 81 is determined by finding a number that multiplies by itself to result in 81. Hence, \( \sqrt{81} = 9 \).

Step 3: Calculate 4 Raised to the Power of 2

The expression \( 4^2 \) means 4 multiplied by itself, so \( 4^2 = 16 \).

Step 4: Combine the Results

Now, substitute the results into the original expression: \( 10 + 9 - 16 \). First, add 10 and 9 to get 19. Then subtract 16 from 19 to find the final answer, which is 3.

Key Concepts

Square Root Calculation

Understanding how to calculate square roots can help you solve math problems with more confidence. A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 100 is denoted as \( \sqrt{100} \). To find this, ask yourself: what number, when multiplied by itself, equals 100? The answer is 10 because \( 10 \times 10 = 100 \). Similarly, for \( \sqrt{81} \), you need a number that results in 81 when squared. This number is 9, since \( 9 \times 9 = 81 \).

• Remember, square roots are positive by convention. So, \( \sqrt{100} = 10 \), not \(-10\).
• Square root symbols are used to denote principal (positive) square roots unless stated otherwise.

Exponentiation

Exponentiation is a fundamental operation in mathematics where a number, known as the base, is multiplied by itself a specific number of times, indicated by the exponent. For example, in the expression \( 4^2 \), 4 is the base, and 2 is the exponent. This means you multiply 4 by itself: \( 4 \times 4 = 16 \). This operation is read as "four squared."

• The exponent indicates how many times the base is used as a factor.
• An important tip is to remember that any number raised to the power of 0 is 1, e.g., \( 4^0 = 1 \).

Exponentiation also follows specific rules, like the product of powers rule, which states that \( a^m \times a^n = a^{m+n} \). This rule applies when multiplying powers with the same base.

Order of Operations

Knowing the order of operations is crucial when solving math expressions involving multiple operations. This rule helps ensure that everyone solves mathematical expressions consistently and accurately.
PEMDAS, an acronym for Parentheses, Exponents, Multiplication and Division (left to right), and Addition and Subtraction (left to right), is helpful in remembering the order.
For example, in the expression \( \sqrt{100} + \sqrt{81} - 4^2 \), you follow these steps:

• First, calculate \( \sqrt{100} \) and \( \sqrt{81} \).
• Then, compute any exponentiation, like \( 4^2 \).
• Finally, perform addition and subtraction from left to right.

This approach ensures you achieve the correct result: \( 10 + 9 - 16 = 3 \). Make sure to always perform calculations in the correct sequence to avoid mistakes.