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Chapter 7: Problem 86

Is \(\sqrt{(2 x+3)^{8}}=(2 x+3)^{4}\) always, sometimes, or never true? Why?

Short Answer

Expert verified
Always true, because the fourth power of any real number is always non-negative.

Step by step solution

01

Understand the Equation

The given equation is \ \ \ \ \(\sqrt{(2 x+3)^{8}}=(2 x+3)^{4}\).\ Review the property of square roots and exponents before proceeding.
02

Apply the Square Root Property

Recall that \ \ \ \ \(\sqrt{a^{2}}=|a|\). Applying this property to the equation, we get: \ \ \ \ \(\sqrt{(2x + 3)^{8}} = \left| (2x + 3)^{4} \right|\).
03

Consider the Absolute Value

The absolute value of \((2x + 3)^{4}\) is different from \((2x + 3)^{4}\) only if \((2x + 3)^{4}\) is negative. Because any real number raised to the fourth power is always non-negative, \( (2x + 3)^{4} \) is always non-negative.
04

Compare the Equations

Since \((2x + 3)^{4}\) is non-negative, the absolute value of \((2x + 3)^{4}\) is \((2x + 3)^{4}\). Therefore, \(\ \ \ \sqrt{(2 x+3)^{8}}=(2 x+3)^{4}\) is always true.

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Key Concepts

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

Exponents
An exponent represents how many times a number (the base) is multiplied by itself. For instance, in \(a^n\), 'a' is the base and 'n' is the exponent. When you raise a number to a power, you multiply the base by itself 'n' times.
For example: \(3^2 = 3 \times 3 = 9\).
Exponents follow specific rules:
  • Product of Powers: \(a^m \times a^n = a^{m+n}\)
  • Power of a Power: \( (a^m)^n = a^{m \times n}\)
  • Zero Exponent: Any non-zero number raised to the power of zero is 1, \(a^0 = 1\)
Understanding exponents helps in simplifying expressions, as seen in the equation \( \sqrt{(2x + 3)^8} = (2x + 3)^4 \) from the exercise. By applying the properties of exponents, solving these types of equations becomes much easier.
Absolute Value
Absolute value refers to the distance of a number from zero on the number line, regardless of direction. It is always a non-negative number. Mathematically, the absolute value of a number 'a' is written as \|a|\.
For example: \|5| = 5\ and \|-5| = 5\.
Key properties of absolute values include:
  • \|a| \geq 0\ for all real numbers 'a'
  • \|a \times b| = |a| \times |b|\
  • \|a + b| \leq \|a| + |b|\
In our exercise, we dealt with the absolute value property: \( \sqrt{a^2} = |a| \). This helped us simplify \( \sqrt{(2x + 3)^8} = |(2x + 3)^4| \). Because any number to the fourth power is non-negative, \( (2x + 3)^4 = |(2x + 3)^4| \).
Real Numbers
Real numbers include all the numbers on the number line, encompassing rational and irrational numbers. They do not include imaginary or complex numbers. Examples include:
  • Rational Numbers: Fractions and integers that can be expressed as a ratio of two integers, like 1/2 or 3.
  • Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers, such as \ \sqrt{2} \ \ and \ \pi \ \
Real numbers are important in many areas of math, including calculus, algebra, and trigonometry. In our exercise, we consider real numbers where the expression \(2x + 3\) is defined and evaluate using properties like exponentiation and absolute value.

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Most popular questions from this chapter

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