Question

Rewrite the expressions in each pair so that they have the same base. a) \(2^{3}\) and \(4^{2}\) b) \(9^{x}\) and 27 c) \(\left(\frac{1}{2}\right)^{2 x}\) and \(\left(\frac{1}{4}\right)^{x-1}\) d) \(\left(\frac{1}{8}\right)^{x-2}\) and \(16^{x}\)

Short answer

a) \(2^{3}\) and \(2^{4}\). b) \(3^{2x}\) and \(3^{3}\). c) \((\frac{1}{2})^{2x}\) and \((\frac{1}{2})^{2x-2}\). d) \((\frac{1}{2})^{3x-6}\) and \((\frac{1}{2})^{-4x}\).

Step-by-step solution

Rewrite the first pair with the same base

Given expressions: \(2^{3}\) and \(4^{2}\). Rewrite 4 as a power of 2. Since \(4 = 2^{2}\), thus \(4^{2} = (2^{2})^{2}\). Apply the power rule: \((a^{m})^{n} = a^{m \times n}\). Therefore, \(4^{2} = 2^{4}\). The expressions are now in the same base: \(2^{3}\) and \(2^{4}\).

Rewrite the second pair with the same base

Given expressions: \(9^{x}\) and 27. Rewrite 9 and 27 as powers of 3. Since \(9 = 3^{2}\) and \(27 = 3^{3}\), substitute these back into the expressions: \(9^{x} = (3^{2})^{x} = 3^{2x}\) and 27 = \(3^{3}\). The expressions are now in the same base: \(3^{2x}\) and \(3^{3}\).

Rewrite the third pair with the same base

Given expressions: \(\left(\frac{1}{2}\right)^{2 x}\) and \(\left(\frac{1}{4}\right)^{x-1}\). Rewrite \(\frac{1}{4}\) as a power of \(\frac{1}{2}\). Since \(\frac{1}{4} = (\frac{1}{2})^{2}\), substitute this back: \(\left(\frac{1}{4}\right)^{x-1} = \left( (\frac{1}{2})^{2} \right)^{x-1} = (\frac{1}{2})^{2x-2}\). The expressions are now in the same base: \((\frac{1}{2})^{2x}\) and \((\frac{1}{2})^{2x-2}\).

Rewrite the fourth pair with the same base

Given expressions: \(\left(\frac{1}{8}\right)^{x-2}\) and \(16^{x}\). Rewrite \(\frac{1}{8}\) as a power of \(\frac{1}{2}\) and 16 as a power of 2. Since \(\frac{1}{8} = (\frac{1}{2})^{3}\) and \(16 = 2^{4}\), substitute these back: \(\left(\frac{1}{8}\right)^{x-2} = \left( (\frac{1}{2})^{3} \right)^{x-2} = (\frac{1}{2})^{3x-6}\) and \(16^{x} = (2^{4})^{x} = 2^{4x}\). Use the fact that \(\frac{1}{2} = 2^{-1}\): \(16^{x} = (2^{4x}) = (2^{-1})^{-4x} = (\frac{1}{2})^{-4x}\). The expressions are now in the same base: \((\frac{1}{2})^{3x-6}\) and \((\frac{1}{2})^{-4x}\).

Key concepts

Expressions with Same Base

When solving problems involving exponents, it is often useful to express terms using the same base. This makes it easier to compare or manipulate the expressions. For instance, if we have the expressions \( 2^{3} \) and \( 4^{2} \), we know that 4 can be rewritten as a power of 2. This results in \(4 = 2^{2}\). Using this, we can rewrite \(4^{2}\) as: \((2^{2})^{2} = 2^{4}\). This way, we have both expressions in terms of base 2: \(2^{3}\) and \(2^{4}\). This process is crucial in mathematics because it simplifies the process of solving equations and comparing expressions.

In another example, consider \( 9^{x} \) and 27. By recognizing that \( 9 = 3^{2} \) and \( 27 = 3^{3} \), we can rewrite the expressions so they share the same base: \((3^2)^x = 3^{2x}\) and \(27 = 3^3\). Now, they both use base 3, making them much easier to handle.

Rewriting Expressions with Exponents

Rewriting exponents involves expressing them in an alternative form that makes calculations easier. This often includes recognizing numbers as powers of a common base. For example, consider the expressions \( \left( \frac{1}{2} \right)^{2x} \) and \( \left( \frac{1}{4} \right)^{x-1} \). To rewrite them with the same base, notice that \( \frac{1}{4} = \left( \frac{1}{2} \right)^{2} \). This gives us:

\[ \left( \frac{1}{4} \right)^{x-1} = \left( \left( \frac{1}{2} \right)^2 \right)^{x-1} = \left( \frac{1}{2} \right)^{2(x-1)} = \left( \frac{1}{2} \right)^{2x-2} \]

Now both expressions share the base \( \frac{1}{2} \): \( \left( \frac{1}{2} \right)^{2x} \) and \( \left( \frac{1}{2} \right)^{2x-2} \). By rewriting exponents like this, we simplify and unify the solutions.

Power Rules in Exponents

The power rule in exponents is a key tool for managing and simplifying expressions. It states that \( (a^m)^n = a^{m \times n} \). This rule allows us to condense or expand expressions involving exponents. For instance, if we have \( \left(2^3\right)^2 \), we can use the power rule to rewrite it as: \( 2^{3 \times 2} = 2^6 \).

Let's look at another example with fractions: \( \left( \frac{1}{8} \right)^{x-2} \). Recognize that \( \frac{1}{8} = \left( \frac{1}{2} \right)^{3} \), we can rewrite it as:\[ \left( \left( \frac{1}{2} \right)^3 \right)^{x-2} = \left( \frac{1}{2} \right)^{3(x-2)} = \left( \frac{1}{2} \right)^{3x-6} \]

Finally, consider a situation with positive and negative exponents: If we can rewrite 16 as \(2^4\), we know that \(16^x = (2^4)^x = 2^{4x}\). By using the equivalence \( \frac{1}{2} = 2^{-1} \), we rewrite 16 as a negative exponent: \( 16^x = \left( \frac{1}{2} \right)^{-4x} \). Thus:\[ \left( \frac{1}{8} \right)^{x-2} = \left( \frac{1}{2} \right)^{3x-6} \text{ and } \left( \frac{1}{2} \right)^{-4x} \]

Utilize the power rule to simplify complex expressions by breaking them down into manageable parts.