Problem 36 - Elementary Algebra

Question

Simplify. $$ \sqrt{25 h^{44}} $$

Step-by-Step Solution

Answer

5h^{22}

Step 1 - Split the square root

Use the property of square roots that states \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \). First, split \( \sqrt{25 h^{44}} \) into two parts: \( \sqrt{25} \) and \( \sqrt{h^{44}} \).

Step 2 - Simplify the square root of a constant

The square root of 25 is 5, since \( 5 \cdot 5 = 25 \). Thus, \( \sqrt{25} \) simplifies to 5.

Step 3 - Simplify the square root of the variable exponent

Next, simplify \( \sqrt{h^{44}} \) using the property \( \sqrt{a^{2n}} = a^n \). Hence, \( \sqrt{h^{44}} = h^{22} \). This is because \(44 \div 2 = 22 \).

Step 4 - Combine the results

Now combine the simplified parts: \( 5 \cdot h^{22} = 5h^{22} \). Therefore, \( \sqrt{25h^{44}} = 5h^{22} \).

Key Concepts

square roots

A square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because \(5 \cdot 5 = 25\). This process of finding square roots is essential in algebra and helps to simplify expressions.
When dealing with square roots in algebra, remember these key properties:

• \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \)
• \( \sqrt{a^2} = a \)

These properties make it easier to work with complex expressions. For instance, in the example \( \sqrt{25 h^{44}} \), we first split it into \( \sqrt{25} \) and \( \sqrt{h^{44}} \). By doing so, we simplify each part separately and then combine them for the final answer.

exponents

Exponents represent repeated multiplication of a number by itself. For instance, \( h^{44} \) means \( h \) multiplied by itself 44 times. Understanding and manipulating exponents is crucial when simplifying algebraic expressions.
Here are some important properties of exponents:

• \( a^m \cdot a^n = a^{m+n} \)
• \( \frac{a^m}{a^n} = a^{m-n} \)
• \( (a^m)^n = a^{m \cdot n} \)

These rules allow us to simplify expressions involving exponents easily. In the exercise, we used the property \( \sqrt{h^{44}} = h^{22} \). This is derived from the rule \( \sqrt{a^{2n}} = a^n \). So, \( h^{44} \) becomes \( h^{22} \) because \( \frac{44}{2} = 22 \).
As you practice more, you'll become adept at recognizing and applying these rules.

algebraic expressions

Algebraic expressions consist of variables, constants, and arithmetic operations. Simplifying these expressions is a major part of algebra.
For example, in the expression \( \sqrt{25 h^{44}} \), we deal with variables (\( h \)) and constants (25). To simplify, we apply properties of square roots and exponents, breaking down the expression into more manageable parts. This step-by-step approach allows us to transform the expression into its simplest form.
Having a good grasp of combining like terms and using mathematical properties helps to simplify algebraic expressions quickly. This skill is foundational for solving more complex algebraic problems.

properties of exponents

Understanding the properties of exponents is essential for simplifying and solving algebraic expressions. Here are some key properties:

• Product of Powers Property: \( a^m \cdot a^n = a^{m+n} \)
• Quotient of Powers Property: \( \frac{a^m}{a^n} = a^{m-n} \)
• Power of a Power Property: \( (a^m)^n = a^{m \cdot n} \)
• Power of a Product Property: \( (ab)^n = a^n \cdot b^n \)
• Power of a Quotient Property: \( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \)

These properties allow us to simplify expressions efficiently. In our exercise \( \sqrt{25 h^{44}} \), we used the property \( \sqrt{a^{2n}} = a^n \), which is related to the power of a power property.
With practice, these properties become intuitive, making it easier to handle complex algebraic operations.