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Magazine Chapter 7: Problem 87
Simplify each expression. $$\sqrt{3 x^{3}} \cdot \sqrt{6 x^{2}}$$
Short Answer
Expert verified
The simplified expression is onumber\[3 \sqrt{2} x^{\frac{5}{2}}\].
Step by step solution
01
Use the Property of Square Roots
Recall that the product of square roots can be combined into a single square root: onumber\[\text{\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}}\]. So, apply this property: onumber\[\sqrt{3 x^3} \cdot \sqrt{6 x^2} = \sqrt{(3 x^3) \cdot (6 x^2)}\].
02
Multiply the Expressions Inside the Square Root
Perform the multiplication inside the square root. Multiply the constants and variables separately: onumber\[\sqrt{3 \cdot 6 \cdot x^3 \cdot x^2} = \sqrt{18 x^{5}}\].
03
Simplify Inside the Square Root
Simplify the expression inside the square root: onumber\[\sqrt{18 x^{5}}\].
04
Factor Inside the Square Root
Break down 18 and the exponent on x to simplify the expression inside the square root. We know 18 can be factored into 9 and 2: onumber\[\sqrt{9 \cdot 2 \cdot x^5}\].
05
Use the Property of Square Roots Again
Using the square root property again: onumber\[\sqrt{9} \cdot \sqrt{2} \cdot \sqrt{x^5}\].
06
Extract Perfect Squares
Evaluate the square roots of the perfect squares and simplify: onumber\[\sqrt{9} = 3\], onumber\[\sqrt{x^5} = x^{\frac{5}{2}}\]. Therefore, the expression becomes onumber\[3 \cdot \sqrt{2} \cdot x^{\frac{5}{2}}\].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Square Roots
Square roots are a fundamental concept in algebra. The square root of a number is a value that, when multiplied by itself, gives the original number. It's denoted by the radical symbol (鈭�). For example,
Be attentive to the properties of square roots since they simplify many expressions.
- 鈭�9 = 3 because 3 * 3 = 9.
- 鈭�16 = 4 because 4 * 4 = 16.
Be attentive to the properties of square roots since they simplify many expressions.
Multiplication of Variables
Multiplication is a basic algebra operation that involves combining numbers and variables. For example, in the expression 3x^3 * 6x^2, you're multiplying numbers (3 and 6) and variables (x^3 and x^2).
Basic multiplication rules include:
Basic multiplication rules include:
- To multiply constants, simply multiply them together: 3 * 6 = 18.
- When multiplying variables with the same base, add their exponents: x^3 * x^2 = x^(3+2) = x^5.
Property of Square Roots
The property of square roots allows you to combine or separate square roots for simplification. A crucial formula is
鈭歔(3x^3) * (6x^2)] = 鈭�(18x^5). Using this property can make complex expressions much more approachable. To separate or 'break down' a square root for further simplification, use:
- 鈭歛 * 鈭歜 = 鈭�(a * b).
鈭歔(3x^3) * (6x^2)] = 鈭�(18x^5). Using this property can make complex expressions much more approachable. To separate or 'break down' a square root for further simplification, use:
- 鈭�(a * b) = 鈭歛 * 鈭歜.
Perfect Squares
Perfect squares are numbers or expressions that are the product of an integer or variable multiplied by itself. Examples of perfect squares include:
鈭�(9x^4 * 2x) = 3x^2 * 鈭�(2x), aid in easy simplification. A thorough understanding of perfect squares ensures quick and precise simplifications.
- 9 (because 3 * 3 = 9).
- 16 (because 4 * 4 = 16).
- x^4 (because (x^2) * (x^2) = x^4).
鈭�(9x^4 * 2x) = 3x^2 * 鈭�(2x), aid in easy simplification. A thorough understanding of perfect squares ensures quick and precise simplifications.
