Problem 27 - College Algebra with Modeling and Visualization

Question

Use positive exponents to rewrite. $$ \sqrt{y \cdot \sqrt{y}} $$

Step-by-Step Solution

Answer

\(y^{3/4}\)

Step 1: Rewrite the square root as an exponent

The square root of a number can be written as a fractional exponent. Thus, for any number \(a\), the square root \(\sqrt{a}\) is written as \(a^{1/2}\). Apply this to the entire expression: \(\sqrt{y \cdot \sqrt{y}} = (y \cdot \sqrt{y})^{1/2}\).

Step 2: Expand the inner square root

The expression inside the main square root is \(y \cdot \sqrt{y}\). The inner square root \(\sqrt{y}\) can be rewritten as \(y^{1/2}\). Thus, this portion becomes \(y \cdot y^{1/2}\).

Step 3: Combine like bases

Recall that when multiplying like bases, you add their exponents. Therefore, \(y \cdot y^{1/2} = y^{1} \cdot y^{1/2} = y^{1 + 1/2} = y^{3/2}\).

Step 4: Apply the outer exponent

Now use the result from Step 3, \(y^{3/2}\), and apply the outer exponent which is \((1/2)\). Applying the rule of raising a power to a power, you multiply the exponents: \((y^{3/2})^{1/2} = y^{(3/2) \cdot (1/2)} = y^{3/4}\).

Key Concepts

Fractional Exponents

Fractional exponents may seem challenging at first, but they are an easier way to express roots. When you see a fraction in an exponent, it indicates a root. For instance, in \(a^{1/2}\), the "1/2" means the square root. This small fraction gives us a big hint about what kind of root we are looking for. Here's why it's useful:

• It turns complicated root expressions into simple power expressions.
• Allows use of the same rules and properties as whole-number exponents.

To rewrite roots as exponents, the numerator is the power, and the denominator is the root. So, \(a^{m/n}\) means take the \(n\)-th root of \(a^m\). Most often, students see these with square roots \((m/n = 1/2)\), but they also show up in cube roots \((m/n = 1/3)\), fourth roots, and more. Notice how this ties into combining exponents later on.

Square Roots

Square roots are one of the most encountered roots. The square root of a number \(a\), denoted as \(\sqrt{a}\), is a value that, when multiplied by itself, gives \(a\). In terms of exponents, this is the same as raising \(a\) to the power of 1/2, or \(a^{1/2}\). Why is this useful?

• Makes calculations with power rules easier.
• Simplifies expressions in algebraic equations.

In our exercise, the term \(\sqrt{y}\) can be rewritten as \(y^{1/2}\). This transition from radical to fractional exponent form is handy because it lets us use exponent rules directly. Remember, the square root is always looking to "undo" the squaring of a number, returning it to its base form.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions using algebraic rules. This skill includes handling exponents, factoring, distributing, and combining like terms. In contexts involving fractional exponents, it primarily means adding, subtracting, or multiplying these expressions correctly.When working with exponents, remember:

• Add exponents when multiplying like bases: \(y^a \cdot y^b = y^{a+b}\).
• Multiply exponents when taking a power of a power: \((y^a)^b = y^{a\cdot b}\).

Our exercise involved both multiplying bases with fractional exponents and using the power of a power rule. The original expression \(y \cdot y^{1/2} = y^{1 + 1/2} = y^{3/2}\) applies the concept of adding exponents. The final step, \((y^{3/2})^{1/2} = y^{3/4}\), uses the multiplication of exponents. These algebraic skills not only simplify mathematical expressions but also enhance problem-solving efficiency.