/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 57 \(\sqrt{81 a^7 b^{12} c^9}\)... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 57

\(\sqrt{81 a^7 b^{12} c^9}\)

Short Answer

Expert verified
The simplified form of \( \sqrt{81 a^7 b^{12} c^9} \) is \( 9a^{3}b^{6}c^{4}\sqrt{a} \)

Step by step solution

01

Break down the given term into its smaller components

Doing this, we get \( \sqrt{81} \times \sqrt{a^7} \times \sqrt{b^{12}} \times \sqrt{c^9} \)
02

Simplify each term individually

This step requires knowledge of square roots. \( \sqrt{81} \) Exponent of 7, 12, 9 can be divided by 2 as they are even. Thus, \(a^3\) still has a radical, and we get: \( 9a^{3}b^{6}c^{4}\sqrt{a} \)
03

Bring the terms together

The final step is to bring all the terms together. So, the solution will be \( 9a^{3}b^{6}c^{4}\sqrt{a} \)

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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 mathematics that allows us to find a number which, when multiplied by itself, gives the original number. For instance, the square root of 81 is 9, because 9 times 9 equals 81. When dealing with square roots in algebraic expressions, we often encounter roots of both numerical and variable components.

To simplify any square root, follow these steps:
  • Identify the square root of the numerical part, if possible. For example, the square root of 81 is straightforward because it鈥檚 a perfect square.
  • For variables, use the rule \( \sqrt{a^n} = a^{n/2} \). If the exponent is odd, you take the largest even number below it for simplification and leave the rest inside the square root.
Understanding and manipulating square roots is crucial, especially when simplifying expressions including square roots, like \( \sqrt{a^7} \), which deals with both whole numbers and remainders.
Exponent Laws
Exponent laws, also known as the laws of exponents or powers, help simplify expressions involving powers of numbers or variables. A critical law to note is that when you take the square root of a power, you're halving the exponent. For example, \( \sqrt{a^n} = a^{n/2} \).

Here are some basic exponent laws:
  • Product of Powers: \( x^a \times x^b = x^{a+b} \)
  • Power of a Power: \( (x^a)^b = x^{a \times b} \)
  • Quotient of Powers: \( \frac{x^a}{x^b} = x^{a-b} \)
  • Zero Exponent: \( x^0 = 1 \) for any \( x eq 0 \)
  • Negative Exponent: \( x^{-a} = \frac{1}{x^a} \)
Applying these laws can simplify complex algebraic expressions, especially when dealing with roots and powers of variables as seen in expressions like \( \sqrt{a^7} \). Using these laws helps to reduce expressions into their simplest forms.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and mathematical operations (addition, subtraction, multiplication, division). Simplifying algebraic expressions involves combining like terms and reducing expressions to their simplest form using rules and properties of numbers and operations.

Consider an expression like \(9a^{3}b^{6}c^{4}\sqrt{a}\). The goal is to simplify wherever possible:
  • Identify and combine like terms, which in this example, is not possible since they are already distinguished terms.
  • Simplify roots and exponents where possible using exponent laws. Here, the large powers of variables \(a, b, c\) have already been simplified using the rule \( \sqrt{a^n} = a^{n/2} \).
Simplifying algebraic expressions helps in understanding and solving equations more efficiently and reveals the true relationships represented by the terms in the expression.

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Let \(g\) be a vertical stretch by a factor of 2 , followed by a translation 2 units up of the graph of \(f(x)=\sqrt{x}+3\).

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