Problem 76 - Elementary Algebra

Question

For the following problems, simplify each expression by removing the radical sign. $$ \sqrt{225 p^{14} r^{16}} $$

Step-by-Step Solution

Answer

Answer: The simplified form is \(15p^7r^8\).

Step 1: Identify each term under the radical

Under the radical, we have three terms: 225, \(p^{14}\), and \(r^{16}\).

Step 2: Find the square root of 225

Since 225 is equal to \(15^2\), the square root of 225 is 15: \(\sqrt{225}=15\).

Step 3: Find the square root of \(p^{14}\)

To find the square root of \(p^{14}\), recall that the square root of a variable raised to an even exponent is the variable raised to half of that exponent: $$\sqrt{p^{14}} = p^{\frac{14}{2}} = p^7$$.

Step 4: Find the square root of \(r^{16}\)

Similarly, the square root of \(r^{16}\) is \(r^{\frac{16}{2}} = r^8\).

Step 5: Combine the square roots

Now we can combine the square roots of each term and simplify the expression: $$\sqrt{225 p^{14} r^{16}} = 15p^7r^8$$. This is the final simplified expression: \(\boxed{15p^7r^8}\).

Key Concepts

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. It is denoted by the radical symbol (√). For example, the square root of 225 is 15 because 15 multiplied by 15 equals 225.
The concept of square roots extends beyond numbers to include algebraic terms. When dealing with variables, like in the expression \(\sqrt{p^{14}}\), the principle remains the same: find a value that, when squared, equals the original expression under the radical.

• Numerical square roots: These involve numbers, e.g., \(\sqrt{225} = 15\) since \(15^2 = 225\).
• Variable square roots: These deal with variables, e.g., \(\sqrt{p^{14}} = p^7\) because \(p^{7} \times p^{7} = p^{14}\).

Knowing how to compute square roots of both numbers and algebraic expressions simplifies the expressions effectively.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. For example, \(p^{14}\) represents \(p\) multiplied by itself 14 times.
When you take a square root of an expression with an exponent, you are essentially halving the exponent. This is because the square root function is the inverse operation of squaring. In mathematical terms, \(\sqrt{x^n} = x^{n/2}\).

• The power of powers rule is applied when simplifying: \(\sqrt{p^{14}} = p^{14/2} = p^7\), as seen in the tearing down of the expression.
• Understanding exponents is key because they show us how many times a number is used as a factor.

Exponents also enable simplification in algebraic expressions and play a crucial role in solving square roots of variable terms.

Algebraic Expressions

Algebraic expressions combine numbers, variables, and operations (like addition, subtraction, multiplication, and division) into mathematical phrases. In the exercise, it involved simplifying the expression \(\sqrt{225 p^{14} r^{16}}\) by dealing with both the numerical and variable components.
The simplification process involves analyzing each component separately to remove any radicals:

• Numerical part: Simplifying \(\sqrt{225}\) to 15.
• Variable parts: \(\sqrt{p^{14}} = p^7\) and \(\sqrt{r^{16}} = r^8\).

After simplifying the parts, they are combined to form the final simplified expression, \(15p^7r^8\).
Mastering algebraic expressions is foundational for solving more complex algebra problems, allowing for clean and concise representations of mathematical ideas. It's the backbone of modern algebra, enabling complex operations to be conducted with simplicity and clarity.