Problem 62 - Algebra 2

Question

The expression \(\sqrt{180 a^{2} b^{8}}\) is equivalent to which of the following? A. 5\(\sqrt{6}|a| b^{4}\) B. 6\(\sqrt{5}|a| b^{4}\) C. 3\(\sqrt{10}|a| b^{4}\) D. 36\(\sqrt{5}|a| b^{4}\)

Step-by-Step Solution

Answer

The equivalent expression is C: 3\(\sqrt{10}|a| b^{4}\).

Step 1: Simplify the square root

First, recognize \[\sqrt{180 a^{2} b^{8}}\] as the square root of the product of several terms. This can be simplified by separating the expression into primes and squares: \[180 = 2 \times 3^2 \times 5\] and recognizing \(a^{2}\) and \(b^{8}\) are perfect squares.

Step 2: Simplify perfect squares

Inside the square root, separate the perfect squares from the non-perfect squares: \[\sqrt{180 a^{2} b^{8}} = \sqrt{(2\times 3^2 \times 5) \times a^2 \times b^8} = \sqrt{3^2 \times a^2 \times b^8} \times \sqrt{2 \times 5}\].

Step 3: Calculate the square of perfect squares

Compute the square root of all perfect squares:\[\sqrt{3^2 \times a^2 \times b^8} = 3 \times |a| \times b^4\].

Step 4: Simplify remaining terms

Now simplify the remaining square root: \[\sqrt{10}\]. Combine all these parts together:\[3 \times |a| \times b^4 \times \sqrt{10} = 3\sqrt{10}|a|b^4\].

Step 5: Compare with given choices

Compare the simplified result, \(3\sqrt{10}|a|b^4\), with the list of choices provided to find the corresponding letter. This expression matches option C.

Key Concepts

Perfect Squares

A perfect square is a number that results from squaring a whole number. For example, 4 is a perfect square because it equals the square of 2 (since 2 x 2 = 4).

When simplifying expressions inside a square root, recognizing perfect squares can be very helpful. This is because perfect squares can be easily taken outside the square root, simplifying the expression.

Consider the expression \(a^2\). It is a perfect square, as it equals \((a)^2\). Similarly, \(b^8\) is a perfect square because it equals \((b^4)^2\). These can be simplified as \(|a|\) and \(b^4\) respectively, when taken out of the square root.

Prime Factorization

Prime factorization is breaking down a number into its basic building blocks: prime numbers. Prime numbers are numbers that are only divisible by 1 and themselves, such as 2, 3, 5, 7, etc.

To simplify a radical expression, you can use prime factorization to identify perfect squares within the number underneath the square root.

For example, in the number 180, we can break it down using prime factorization as follows:

• 180 can be divided by 2 (the smallest prime number), giving 90.
• 90 can be divided by 2 again, giving 45.
• 45 can be divided by 3, giving 15.
• 15 can be divided by 3, giving 5.
• The primes that make up 180 are thus 2, 3, 3, and 5.

Using these, we can reorder 180 as \(2 \times 3^2 \times 5\), which helps in further simplifying the square root.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. The square root is often denoted by the radical symbol \(\sqrt{}\).

For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).

When simplifying expressions that involve square roots, the goal is to break down the expression so that any perfect squares can be fully simplified. This often involves moving numbers or variables out from under the square root when possible.

In our expression \(\sqrt{180 a^{2} b^{8}}\), perfect squares \(3^2\), \(a^2\), and \(b^8\) can be taken out of the radical sign to simplify the expression further:

• The \(3^2\) results in pulling out a 3.
• The \(a^2\) results in \(|a|\).
• The \(b^8\) results in pulling out \(b^4\).

After removing these perfect squares, you are left with \(\sqrt{10}\), the remaining product underneath the radical that cannot be simplified further.