Problem 16 - Algebra 1

Question

Use the quotient of powers property to simplify the expression. $$ \left(\frac{x^{4}}{2^{3}}\right)^{2} $$

Step-by-Step Solution

Answer

\(\frac{x^8}{64}\)

Step 1: Identify the base and the exponent

In the given expression \(\left(\frac{x^{4}}{2^{3}}\right)^{2}\), we can see that the base is \(\frac{x^{4}}{2^{3}}\) and the exponent is 2.

Step 2: Apply the power of a quotient rule

By applying the power of a quotient rule, the given expression simplifies to \(\frac{(x^{4})^2}{(2^{3})^2}\).

Step 3: Simplify the expression

Simplifying the expression further gives us \(\frac{x^{4*2}}{2^{3*2}} = \frac{x^8}{2^6} = \frac{x^8}{64}\).

Key Concepts

Simplifying Expressions

The process of simplifying expressions is essential in mathematics as it helps in making complex equations more understandable and manageable. In essence, simplifying an expression means reducing it to its simplest form, while maintaining its original values. This involves combining like terms, eliminating unnecessary parentheses, and using properties of arithmetic to condense the expression as much as possible.

Consider the exercise \(\left(\frac{x^{4}}{2^{3}}\right)^{2}\). To simplify this, we must first look for any common factors, properties of exponents, or arithmetic operations that can be applied to condense the expression. In this case, we leverage the properties of exponents to transform the expression into a simpler form.

Power of a Quotient Rule

When it comes to simplifying expressions involving powers and quotients, the power of a quotient rule is a powerful tool. This rule states that an entire fraction raised to an exponent is equal to each part of the fraction raised to that exponent.

Formally, the rule can be expressed as \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\) for any non-zero number \(b\) and any real number \(n\). Applying this rule helps break down a complex fraction into a simpler form, making the rest of the simplification process more straightforward.

Returning to our exercise \(\left(\frac{x^{4}}{2^{3}}\right)^{2}\), by applying the power of a quotient rule, the exponent is distributed to both the numerator and the denominator resulting in \(\frac{(x^{4})^2}{(2^{3})^2}\). This step was pivotal in the simplification process of the given exercise.

Exponents

Exponents are repeated multiplications of the same factor. The number of times the factor is multiplied by itself is indicated by the exponent. For example, \(x^3\) means \(x\) multiplied by itself three times (\(x*x*x\)).

Understanding how to manipulate exponents is crucial when working with algebraic expressions. There are several properties of exponents that can help simplify expressions:

• The Product of Powers property: \(x^n \times x^m = x^{n+m}\)
• The Quotient of Powers property: \(\frac{x^n}{x^m} = x^{n-m}\), when \(n > m\)
• The Power of a Power property: \(\(x^n\)^m = x^{n\times m}\)

In the given exercise, we used the concept of raising a power to a power, where the exponents are multiplied together. This turned \(x^{4*2}\) into \(x^8\) and \(2^{3*2}\) into \(2^6\), or 64, which illustrates the simplification of an exponential expression.