Problem 30 - Intermediate Algebra

Question

Simplify each expression. Assume that the variables can be any real number, and use absolute value symbols See Example 2. $$ \left(25 a^{8}\right)^{1 / 2} $$

Step-by-Step Solution

Answer

The simplified expression of \( \left(25a^8\right)^{1/2} \) is \(5a^4\).

Step 1: Understand the Expression

We need to simplify the expression \( \left(25a^8\right)^{1/2} \). The expression involves taking the square root (or the 1/2 power) of the product \( 25a^8 \).

Step 2: Simplify the Square Root of a Product

Apply the property of exponents that states \((xy)^n = x^n y^n\) to rewrite the expression as \((25)^{1/2} (a^8)^{1/2}\).

Step 3: Calculate the Square Root of 25

Since \(25\) is a perfect square, its square root is \(5\). Thus, \((25)^{1/2} = 5\).

Step 4: Simplify the Square Root of a Power

For the term \((a^8)^{1/2}\), use the rule \((x^m)^n = x^{mn}\). Here, \(m=8\) and \(n=1/2\), so \((a^8)^{1/2} = a^{8 \times 1/2} = a^4\).

Step 5: Combine the Results

Putting it all together, the expression simplifies to \(5a^4\).

Key Concepts

Properties of Exponents

Exponents are a shorthand way to express repeated multiplication. For example, in the expression \(a^8\), the base \(a\) is multiplied by itself 8 times. Knowing the properties of exponents makes simplifying expressions much easier. Here are some key properties:

• Product of Powers: When multiplying like bases, add the exponents. For example, \(x^m \times x^n = x^{m+n}\).
• Power of a Power: To raise a power to another power, multiply the exponents. So, \((x^m)^n = x^{mn}\).
• Power of a Product: To find the power of a product, apply the exponent to each factor. For example, \((xy)^n = x^n y^n\).

Understanding these properties can help you break down complex expressions, making them simpler to work with. This foundational knowledge supports solving problems like simplifying \((25 a^8)^{1/2}\) by applying exponent rules accurately.

Square Roots of Real Numbers

A square root finds the value that, when multiplied by itself, gives the original number. The square root of a perfect square is straightforward because it results in an integer. For example, the square root of 25 is 5. This is because \(5 \times 5 = 25\).When dealing with variables, square roots can involve applying the concept to powers. For instance, the square root \((a^8)^{1/2}\) requires using the property of exponents that says \((x^m)^{1/2} = x^{m/2}\).Remember:

• The square root of a number and the square root of its power can often be rewritten using fractional exponents, like \((25)^{1/2}\) or \((a^8)^{1/2}\), simplifying the computation.
• Variables can represent positive or negative values, hence sometimes absolute values are used when taking roots to ensure the result is non-negative.

Applying these principles helps in simplifying expressions like \((25 a^8)^{1/2}\) into more manageable terms.

Exponent Rules

When simplifying expressions with exponents, understanding the rules governing their manipulation is crucial. Here are some important exponent rules:

• Zero Exponent Rule: Any non-zero base raised to the power of zero is 1. For example, \(x^0 = 1\) for \(x eq 0\).
• Negative Exponent Rule: A negative exponent means the reciprocal of the base raised to the opposite positive exponent, such as \(x^{-n} = \frac{1}{x^n}\).
• Fractional Exponents: A fractional exponent like \(x^{m/n}\) signifies a root. The denominator is the root, and the numerator is the power. For example, \(x^{1/2}\) is the square root of \(x\), and \(x^{3/2}\) is the same as the square root of \(x^3\).

In simplifying the expression \((25 a^8)^{1/2}\), the power of a product is applied by considering \((25)^{1/2} \cdot (a^8)^{1/2}\), resulting in a further breakdown into \(5 a^4\). Mastery of these rules allows you to approach such expressions with confidence and ease.