Problem 90 - Intermediate Algebra

Question

Evaluate each expression. See Example \(9 .\) $$ \frac{(6-5)^{4}+21}{27-(\sqrt{16})^{2}} $$

Step-by-Step Solution

Answer

The expression evaluates to 2.

Step 1: Simplify the Numerator

First, evaluate the subtraction inside the parentheses: \[ 6 - 5 = 1 \]Next, raise the result to the power of 4: \[ 1^4 = 1 \]Then, add 21 to this result: \[ 1 + 21 = 22 \]So, the numerator simplifies to 22.

Step 2: Simplify the Denominator

Begin by evaluating the square root: \[ \sqrt{16} = 4 \]Then, raise this result to the power of 2: \[ 4^2 = 16 \]Now, subtract this from 27: \[ 27 - 16 = 11 \]Thus, the denominator simplifies to 11.

Step 3: Divide the Simplified Numerator by Denominator

Now, divide the simplified numerator by the simplified denominator: \[ \frac{22}{11} = 2 \]This gives you the value of the entire expression.

Key Concepts

Order of Operations

The order of operations is a set of rules that determines the sequence in which different operations in a mathematical expression are carried out. It's crucial to follow these rules to arrive at the correct answer. Oftentimes, the acronym PEMDAS is used to remember the order:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the given problem, the expression \[\frac{(6-5)^{4}+21}{27-(\sqrt{16})^{2}} \] illustrates the correct order to apply these operations. First, solve anything inside parentheses. Next, handle any exponents or square roots. Finally, perform any addition or subtraction. By following this sequence, you ensure that every part of the expression is simplified correctly.

Exponents

Exponents represent repeated multiplication of a number by itself. If you have an expression like \[x^n\], the "base" is \(x\), and \(n\) is the "exponent." It means that \(x\) is multiplied by itself \(n\) times. For example, \[5^3 = 5 \times 5 \times 5 = 125\].
In the exercise, we see an exponent in the numerator: \[(6-5)^4\]. Here, after evaluating the expression inside the parentheses, which simplifies to \(1\), you then raise it to the 4th power: \[1^4 = 1\]. Understanding exponents helps in recognizing how quickly numbers can grow and is essential in correctly simplifying and calculating expressions involving powers.

Square Roots

A square root is essentially the opposite of squaring a number. It answers the question: "What number, when multiplied by itself, gives me \(x\)?" The square root symbol \(\sqrt{}\) indicates this operation. For instance, \[\sqrt{16} = 4\] because \[4 \times 4 = 16\].
In the given problem, the denominator contains \[(\sqrt{16})^2\]. First, calculate the square root of 16, which is 4. Then, you are asked to square that result, effectively returning to the original number: \[4^2 = 16\]. This back and forth simplification emphasizes the relationship between squaring a number and taking a square root, and highlights how these operations are crucial in solving algebraic expressions efficiently.