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Magazine Chapter 8: Problem 53
Simplify. $$\sqrt{21 n^{5} p}$$
Short Answer
Expert verified
n^{2} \times \sqrt{21np}
Step by step solution
01
Identify the parts under the square root
Determine the components inside the square root. Here, we have 21, \(n^{5}\), and \(p\).
02
Break down the components with exponents
Recognize that \(n^{5}\) can be factored to \(n^{4} \times n\). We do this because \(n^{4}\) is a perfect square.
03
Separate the components
Express the square root as \(\sqrt{21} \times \sqrt{n^{4}} \times \sqrt{n} \times \sqrt{p}\).
04
Simplify the perfect squares
\(\sqrt{n^{4}} = n^{2}\) because the square root of a perfect square is just half the exponent.
05
Combine the simplified parts
Combine the simplified components: \(n^{2} \times \sqrt{21} \times \sqrt{n} \times \sqrt{p}\).
06
Reorganize the expression
Reorganize to put all the square roots together: \(n^{2} \times \sqrt{21np}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Square Roots
Understanding square roots is vital in algebra. A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 \times 3 = 9. In algebra, square roots can simplify complex expressions.
To simplify \(\text{鈭殅 21 n^{5} p\), we need to break down the expression into manageable parts. Each part under the square root can be simplified individually and then recombined. For instance, 鈭�21 can be left as it is if it鈥檚 not a perfect square, and 鈭歯^5 can be split by recognizing parts that are perfect squares.
To simplify \(\text{鈭殅 21 n^{5} p\), we need to break down the expression into manageable parts. Each part under the square root can be simplified individually and then recombined. For instance, 鈭�21 can be left as it is if it鈥檚 not a perfect square, and 鈭歯^5 can be split by recognizing parts that are perfect squares.
Exponents
Exponents represent repeated multiplication. For instance, $$n^{5}$$ means n multiplied by itself 5 times: $$n \times n \times n \times n \times n$$. When simplifying expressions with square roots, exponents play a crucial role.
We can break down exponents to simplify square roots easier. In our example, $$n^{5} = n^{4} \times n$$. This is useful because $$n^{4}$$ is a perfect square and can be further simplified.
Understanding exponents helps in breaking down complex expressions into simpler parts. This makes further simplifications easier and clearer.
We can break down exponents to simplify square roots easier. In our example, $$n^{5} = n^{4} \times n$$. This is useful because $$n^{4}$$ is a perfect square and can be further simplified.
Understanding exponents helps in breaking down complex expressions into simpler parts. This makes further simplifications easier and clearer.
Perfect Squares
A perfect square is a number that can be expressed as the square of an integer. Examples include 16 (4 \times 4 = 16), 9 (3 \times 3 = 9), and 25 (5 \times 5 = 25). In algebra, perfect squares make simplification easier, especially under square roots.
In our equation, $$n^4$$ is a perfect square because it is $$(n^2)^2$$. Recognizing perfect squares under the square root allows us to simplify: \( \text{鈭殅 n^{4} = n^{2} \). This reduces the complexity of the expression.
In our equation, $$n^4$$ is a perfect square because it is $$(n^2)^2$$. Recognizing perfect squares under the square root allows us to simplify: \( \text{鈭殅 n^{4} = n^{2} \). This reduces the complexity of the expression.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions using algebraic rules. It helps in transforming a complex expression into a simpler or more usable form.
For example, with our expression \(\text{鈭殅 21 n^{5} p\), we use algebraic manipulation to simplify it step by step:
For example, with our expression \(\text{鈭殅 21 n^{5} p\), we use algebraic manipulation to simplify it step by step:
- First, break down exponents: $$n^{5} = n^{4} \times n$$
- Next, separate the expression: $$\text{鈭殅 21 \times \text{鈭殅 n^{4} \times \text{鈭殅 n \times \text{鈭殅 p$$
- We then simplify the perfect squares: $$\text{鈭殅 n^{4} = n^{2}$$
- Finally, combine the simplified parts: $$n^{2} \times \text{鈭殅 21 n p$$
