Problem 30 - Elementary Algebra

Question

For the following problems, simplify each of the radical expressions. $$ \sqrt{64 a^{7} b^{3}} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(\sqrt{64 a^7 b^3}\). Answer: \(8a^3b\)

Step 1: Simplify the numerical part

First, find the square root of 64, which is 8.

Step 2: Simplify the variable part with even exponent

Next, find the square root of \(a^{7}\). Since the exponent is odd, we can round down to the nearest integer: $$\sqrt{a^7} = a^{\lfloor \frac{7}{2} \rfloor} = a^3.$$

Step 3: Simplify the variable part with odd exponent

Now, find the square root of \(b^{3}\). Since the exponent is odd, we can round down to the nearest integer: $$\sqrt{b^3} = b^{\lfloor \frac{3}{2} \rfloor} = b^1 = b.$$

Step 4: Write down the simplified expression

Combine the results from Steps 1, 2, and 3 to write the simplified expression: $$\sqrt{64 a^7 b^3} = 8a^3b.$$

Key Concepts

Understanding Square Roots

When we talk about square roots, we're referring to finding a number which, when multiplied by itself, gives the original number. In simple terms, if you take the square root of 64, you are looking for a number that, when squared (multiplied by itself), results in 64. That number is 8.

• The square root of a perfect square such as 64 is always a whole number. This makes simplifying such expressions straightforward.
• It’s essential to recognize perfect squares quickly, like 4, 9, 16, 25, which simplifies computations.

Remember, the square root function applies to both numbers and variables and is usually symbolized as \( \sqrt{} \), where what's inside the symbol is called the radicand.

Dealing with Even and Odd Exponents

Exponents in expressions require special attention, especially when simplifying under a square root.
When dealing with even exponents, the process is more straightforward because you can evenly divide them by two. For example, for an exponent like \(a^6\), the square root would be \(a^3\) because \(6/2 = 3\).
However, for odd exponents, it's a bit different. With \(a^7\), you’re looking at bending the exponent rules a bit by dividing, but since it's odd, you get a remainder. Thus, \( \lfloor 7/2 \rfloor = 3 \), meaning we take \(a^3\) out of the square root and leave \(a\) inside, as reflected in expressions like \(\sqrt{a^7} = a^3\cdot \sqrt{a}\).
This approach helps in simplifying expressions further, especially when variable terms have both even and odd exponents.

Simplifying Expressions Effectively

Simplifying radical expressions is about breaking them into a simpler, more manageable form without changing their value. It often involves working with numerical and variable parts separately.

• First, simplify the numerical part by finding and factoring out perfect squares.
• Then, for variables, extract factors from the square root with even exponents, significantly reducing complexity.
• Recombine all simplified parts to form the final expression.

For instance, with the problem \(\sqrt{64 a^7 b^3}\), we see 64 simplifies to 8, \(a^7\) simplifies to \(a^3\cdot \sqrt{a}\), and \(b^3\) to \(b^1\cdot \sqrt{b}\). Bringing these simplified components together gives \(8a^3b\), a much cleaner and more understandable form.
Staying systematic and breaking down the problem into smaller pieces ensures that you simplify expressions correctly and efficiently.