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Chapter 7: Problem 88

Simplify. $$(5 a \sqrt{a b})^{2}$$

Short Answer

Expert verified
The simplified expression is \( 25a^{3}b \).

Step by step solution

01

Understand the problem

We need to simplify the expression \((5a \sqrt{ab})^{2}\). This means we need to expand and simplify the given expression.
02

Apply the exponent

Rewrite the expression as \((5a \sqrt{ab})^{2} = (5a \sqrt{ab}) \times (5a \sqrt{ab})\).
03

Distribute the terms inside the radical

Since we have \( \sqrt{ab} \) inside the parenthesis, we can use the property \(( \sqrt{x}) \times ( \sqrt{x}) = x\). Therefore, \( \sqrt{ab} \times \sqrt{ab} = ab\).
04

Calculate the product

Combine all the factors: \((5a)^{2} \) and \(( \sqrt{ab})^{2} = ab\). So, \( (5a \sqrt{ab}) \times (5a \sqrt{ab}) = 25a^{2} \times ab = 25a^{3}b \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

algebraic expressions
Algebraic expressions involve numbers, variables, and arithmetic operations. In the exercise, we deal with an expression containing variables and coefficients. Simplifying these expressions helps in solving algebra problems more efficiently. The given problem is \( (5a \sqrt{ab})^{2} \), which is an algebraic expression.
It involves multiplication and exponentiation of variables.
exponents
Exponents indicate how many times a number or variable is multiplied by itself. In our problem, the exponent is 2, meaning the expression inside the parentheses needs to be multiplied by itself.
For the expression \( (5a \sqrt{ab})^{2} \), this works out as \( (5a \sqrt{ab}) \times (5a \sqrt{ab}) \). Understanding exponents is essential in algebra.
When simplifying, remember these key properties:
  • \
radicals
Radicals involve the root of a number or variable. Here, \( \sqrt{ab} \) is a radical expression. Simplifying radicals means rewriting them without the root, if possible. In the exercise, use the property \( \sqrt{x} \times \sqrt{x} = x \).
This translates \sqrt{ab} \times \sqrt{ab} into ab, eliminating the radical. Understanding and simplifying radicals is crucial for algebraic simplification.
These math tricks are very helpful as they make complicated calculations easier.
After simplifying \( 5a \sqrt{ab} \times 5a \sqrt{ab}\), we get the final simplified form: 25a^{3}b.\
Mastering these skills takes practice but significantly aids in algebra studies.

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Most popular questions from this chapter

Because 3 is the square of \(\sqrt{3}\), a binomial such as \(y^{2}-3\) is a difference of two squares. a) Factor \(y^{2}-3\) and \(2 a^{2}-7\) using radicals. b) Use factoring with radicals to solve the equation \(x^{2}-8=0\). c) Assuming \(a\) is a positive real number, solve the equation \(x^{2}-a=0\).

Solve each equation. $$\sqrt{x^{2}-3 x}=x$$

Simplify. $$4 \sqrt{a}(a+\sqrt{a})$$

Solve each equation. $$3+\sqrt{x^{2}-8 x}=0$$

Solve each equation. $$(w+3)^{-1 / 3}=\frac{1}{3}$$
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