Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 88
Replace the symbol \(\square\) with elther \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your answer. \(\sqrt{a^{r}} \square(\sqrt{a})\)
Short Answer
Expert verified
Replace with \( = \) if \( r = 1 \); otherwise, use \( \neq \).
Step by step solution
01
Understanding the Expression
The given expression is \( \sqrt{a^r} \square (\sqrt{a}) \). We need to determine whether to replace \( \square \) with \( = \) or \( eq \) under given conditions for \( a \) and \( r \). The expression involves comparing two square roots.
02
Simplifying the Square Roots
To simplify \( \sqrt{a^r} \), note that \( \sqrt{a^r} = (a^r)^{1/2} = a^{r/2} \). Similarly, \( \sqrt{a} \) can be expressed as \( a^{1/2} \). The comparison becomes \( a^{r/2} \square a^{1/2} \).
03
Setting Conditions for Equality
For \( a^{r/2} = a^{1/2} \) to hold, the exponents must be identical. Thus, we require \( r/2 = 1/2 \), which simplifies to \( r = 1 \). If \( r = 1 \), the expression is true under all conditions where both sides are defined.
04
Setting Conditions for Inequality
If \( r eq 1 \), then \( a^{r/2} \) does not equal \( a^{1/2} \), making \( \sqrt{a^r} eq \sqrt{a} \). Here, inequality holds when \( r eq 1 \).
05
Conclusion
The expression \( \sqrt{a^r} = \sqrt{a} \) is valid if and only if \( r = 1 \). If \( r eq 1 \), the expression becomes \( \sqrt{a^r} eq \sqrt{a} \). Thus, the correct symbol depends on the value of \( r \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Square Roots
The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 times 3 equals 9. When dealing with expressions involving square roots, such as \( \sqrt{a} \), our focus is on finding a number that, when squared, equals \( a \).
Square roots have particular rules that need to be followed:
Square roots have particular rules that need to be followed:
- The square root of a product can be broken into the product of the square roots, \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \).
- The square root of a fraction is the fraction of the square roots, \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).
- Every positive number has two square roots: one positive and one negative, though in much mathematical practice, the positive root is assumed.
Exponents
Exponents, also known as powers, are a way to express repeated multiplication of a number by itself. For example, \( a^3 \) means \( a \times a \times a \). In general, \( a^n \) indicates that \( a \) is multiplied by itself \( n \) times.
When working with exponents, several important rules can be useful:
When working with exponents, several important rules can be useful:
- Multiplying powers with the same base: \( a^m \cdot a^n = a^{m+n} \).
- Dividing powers with the same base: \( \frac{a^m}{a^n} = a^{m-n} \).
- Raising a power to another power: \( (a^m)^n = a^{m\cdot n} \).
- A power of zero: \( a^0 = 1 \) (as long as \( a eq 0 \)).
Inequality
An inequality is a mathematical sentence that shows the relationship between quantities that are not equivalent. For example, \( a < b \) indicates that \( a \) is less than \( b \).
There are several signs used in inequalities:
There are several signs used in inequalities:
- \(<\): less than
- \(>\): greater than
- \(\leq\): less than or equal to
- \(\geq\): greater than or equal to
- \(eq\): not equal to
Equation Solving
Solving equations involves finding the values of the variable that make the equation true. In our given expression \( \sqrt{a^r} \square \sqrt{a} \), determining the right symbol (either \( = \) or \( eq \)) necessitates solving for \( r \).
The steps to solve such an equation include:
The steps to solve such an equation include:
- Identify the expression or equation.
- Simplify each side of the equation if necessary. With \( \sqrt{a^r} = a^{r/2} \), we simplify using exponent rules.
- Check if you can equate the expressions by finding a common base or power.
- Solve for the unknown by balancing the equation.
