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Chapter 9: Problem 30

Find each of the following products. $$ \sqrt{x} \sqrt{x} $$

Short Answer

Expert verified
Answer: The product of the expression \(\sqrt{x}\sqrt{x}\) is \(x\).

Step by step solution

01

Identify the terms to be multiplied

We are given the expression \(\sqrt{x}\sqrt{x}\). Here, the terms to be multiplied are \(\sqrt{x}\) and \(\sqrt{x}\).
02

Multiply the square roots

To multiply the two square roots, we write it like this: $$ \sqrt{x} \cdot \sqrt{x} $$
03

Apply the property of square roots

We can now apply the property of square roots, which states that \(\sqrt{a} \cdot \sqrt{a} = a\). In this case, our "a" is "x", so we have: $$ \sqrt{x} \cdot \sqrt{x} = x $$
04

State the final answer

By multiplying the two square roots together, we have found the product to be: $$ \sqrt{x} \cdot \sqrt{x} = x $$

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Key Concepts

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

Multiplication of Square Roots
When you have square roots being multiplied together, like \(\sqrt{x}\) and \(\sqrt{x}\), it's a straightforward process. Multiplying square roots involves using a property that simplifies multiplication under the radical. If you multiply \(\sqrt{a}\) by \(\sqrt{b}\), you multiply the numbers under the radicals first. This is expressed as:
  • \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\)
  • In simpler cases, such as \(\sqrt{x} \cdot \sqrt{x}\), the solution turns into \(\sqrt{x^2}\).
Keep in mind, \(\sqrt{x} \cdot \sqrt{x}\) simplifies fully into just \(x\) because \(\sqrt{x^2} = x\). This understanding of multiplication makes it easier to manage square roots in algebraic expressions.
Practicing these concepts can build confidence in solving more complex equations involving radicals.
Properties of Square Roots
Square roots have special mathematical properties that simplify how we work with them. One of the main properties helpful in today's exercise is how multiplying square roots together often results in removing the radical. Let's break this property down:
  • \((\sqrt{a})^2 = a\): This tells us that whenever you multiply a square root by itself, you get the number under the square root.
  • Importantly, \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\) allows combining square roots during multiplication.
This comes in handy when simplifying expressions, as seen with \(\sqrt{x} \cdot \sqrt{x}\) simply equaling \(x\).
Recognizing these properties allows you to efficiently handle and work through problems involving square roots. Comprehending them can be a game-changer when simplifying expressions.
Basic Algebra
Basic algebra often serves as the stepping stone for understanding more complex mathematical problems, including dealing with square roots. Algebra is concerned with symbols and the rules for manipulating these symbols.
  • In algebra, terms like variables are used instead of known numbers. For example, "x" is a common variable.
  • When dealing with expressions like \(\sqrt{x} \cdot \sqrt{x}\), the algebraic rules we apply come directly from understanding mathematical properties, like those of square roots.
Performing operations such as multiplication on algebraic terms follows certain principles, as you apply properties rigorously across different problems.
Understanding algebra basics can empower you to simplify and solve expressions smoothly. It is foundational knowledge necessary for dealing with expressions that involve radical terms.

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Most popular questions from this chapter

Simplify each expression by performing the indicated operation. $$ (3-\sqrt{2})(4-\sqrt{2}) $$

Simplify each expression by performing the indicated operation. $$ (\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) $$

Simplify each expression by performing the indicated operation. $$ \frac{1}{4-\sqrt{3}} $$

At a small business, the monthly number of sales \(S\) is approximately related to the number of employees \(E\) by \(S=140+8 \sqrt{E-2}\) (a) Determine the approximate number of sales if the number of employees is 27 . (b) Determine the approximate number of employees if the monthly sales are 268 .

The resonance frequency \(f\) in an electronic circuit containing inductance \(L\) and capacitance \(C\) in series is given by \(f=\frac{1}{2 \pi \sqrt{L C}}\) (a) Determine the resonance frequency in an electronic circuit if the inductance is 9 and the capacitance is 0.0001 . Use \(\pi=3.14\). (b) Determine the inductance in an electric circuit if the resonance frequency is 5.308 and the capacitance is 0.0001 . Use \(\pi=3.14\).
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