Problem 76 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{(a-7)^{8}} $$

Step-by-Step Solution

Answer

Answer: The simplified form of the expression is: $$(a-7)^{4}$$

Step 1: Identify the power inside the square root

In this expression, we are given the power inside the square root as: \((a-7)^{8}\). Notice that the power inside the square root is 8.

Step 2: Apply the rule to simplify the square root

Now, we will apply the rule mentioned in the analysis, where the square root of a number raised to a power is equal to that number raised to half of the power. So, we have: $$ \sqrt{(a-7)^{8}} = (a-7)^{\frac{8}{2}} $$

Step 3: Simplify the exponent and the expression

Finally, we need to simplify the exponent: $$ (a-7)^{\frac{8}{2}} = (a-7)^{4} $$ So, the simplified expression is: $$ \sqrt{(a-7)^{8}} = (a-7)^{4} $$

Key Concepts

square root

A square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 since 3 multiplied by 3 is 9. The symbol for the square root is \( \sqrt{} \). Square roots are used when you need to "undo" a square, which is why square roots and exponents are closely related.
In our exercise, we have a square root of an expression raised to a power: \( \sqrt{(a-7)^8} \). The goal here is to simplify this complex expression into an easier-to-manage form.
Getting familiar with how square roots interact with other mathematical operations is crucial for simplifying expressions.

exponents

Exponents are a way to show that a number, known as the base, is multiplied by itself a specific number of times. For example, in \( 2^3 \), 2 is the base and 3 is the exponent, meaning \( 2 \times 2 \times 2 \). Exponents are fundamental in algebra as they allow us to write repeated multiplication in a concise form.
In the given problem, \( (a-7)^8 \), the expression \((a-7)\) is the base and 8 is the exponent. This symbolizes that the expression \( (a-7) \) is being multiplied by itself 8 times. Applying exponents correctly is crucial because they enable us to both expand and compress mathematical expressions, making them easier to work with.

rules of exponents

The rules of exponents are guidelines that help simplify expressions involving powers. These include:

• \( x^a \times x^b = x^{a+b} \): When multiplying like bases, add their exponents.
• \( \frac{x^a}{x^b} = x^{a-b} \): When dividing like bases, subtract their exponents.
• \( (x^a)^b = x^{a\times b} \): When raising a power to another power, multiply the exponents.

These rules are helpful in our problem where we simplify a square root involving an exponent. The rule \((x^a)^{1/2} = x^{a/2}\) was used to simplify \( \sqrt{(a-7)^8} = (a-7)^{8/2} = (a-7)^4 \). Understanding these rules helps in efficiently breaking down and simplifying complex expressions without losing any mathematical integrity.