Problem 78
Question
Simplify each expression, if possible. All variables represent positive real numbers. $$ \sqrt{y^{5}}-\sqrt{9 y^{5}}-\sqrt{25 y^{5}} $$
Step-by-Step Solution
Verified Answer
The expression simplifies to \(-7y^{2.5}\).
1Step 1: Simplify Each Square Root
First, simplify the expression inside each square root separately. We can use the property \( \sqrt{a^2} = a \) for each case, considering that\( a > 0 \).- For \( \sqrt{y^5} \), notice that: \( y^5 = y^2 \cdot y^2 \cdot y^1 = (y^2)^2 \cdot y = (y^{2.5})^2 \), hence \( \sqrt{y^5} = y^{2.5} \).- For \( \sqrt{9y^5} \), simplify as follows: \( \sqrt{9y^5} = \sqrt{9} \cdot \sqrt{y^5} = 3 \cdot y^{2.5} = 3y^{2.5} \).- For \( \sqrt{25y^5} \), simplify similarly: \( \sqrt{25y^5} = \sqrt{25} \cdot \sqrt{y^5} = 5 \cdot y^{2.5} = 5y^{2.5} \).
2Step 2: Calculate the Overall Expression
Now that each square root has been simplified, plug them back into the overall expression:\( \sqrt{y^5} - \sqrt{9y^5} - \sqrt{25y^5} = y^{2.5} - 3y^{2.5} - 5y^{2.5} \).
3Step 3: Combine Like Terms
Combine the like terms in the expression. All terms are multiples of \( y^{2.5} \):\( y^{2.5} - 3y^{2.5} - 5y^{2.5} = (1 - 3 - 5)y^{2.5} = (-7)y^{2.5} \).
Key Concepts
Square RootsExponentsReal NumbersLike Terms
Square Roots
Square roots are a fundamental concept in algebra which involves finding a number that, when multiplied by itself, gives the original number. The symbol used for the square root is \( \sqrt{} \). For example, \( \sqrt{9} = 3 \) since \( 3 \times 3 = 9 \).
When it comes to expressions with variables, such as \( \sqrt{y^5} \), the process is similar. What we essentially do in this situation is look for pairs of terms that when squared will yield the original term inside the root. For instance, breaking \( y^5 \) into \( y^2 \cdot y^2 \cdot y \), we notice \( y^2 \cdot y^2 = (y^2)^2 \). This can be further simplified to \( y^{2.5} \) because \( (y^{2.5})^2 = y^5 \).
This process of extracting pairs is crucial for simplifying complex expressions involving square roots, as seen in the original exercise.
When it comes to expressions with variables, such as \( \sqrt{y^5} \), the process is similar. What we essentially do in this situation is look for pairs of terms that when squared will yield the original term inside the root. For instance, breaking \( y^5 \) into \( y^2 \cdot y^2 \cdot y \), we notice \( y^2 \cdot y^2 = (y^2)^2 \). This can be further simplified to \( y^{2.5} \) because \( (y^{2.5})^2 = y^5 \).
This process of extracting pairs is crucial for simplifying complex expressions involving square roots, as seen in the original exercise.
Exponents
Exponents are a way to express repeated multiplication of the same number or variable. The exponent tells us how many times the base is used as a factor. For example, in the expression \( 2^3 \), the number 2 is multiplied by itself 3 times (\( 2 \times 2 \times 2 \)).
When dealing with variables raised to exponents, such as \( y^5 \), this means \( y \) is multiplied by itself 5 times. When simplified within square roots, exponents need extra care. We used properties of exponents like \( (a^{m})^n = a^{m\cdot n} \) to break down and simplify complex expressions within roots. So, \( \sqrt{y^5} \) simplified becomes \( y^{2.5} \), using expression \( (y^{2.5})^2 = y^5 \).
Understanding how to manipulate exponents is key in simplification tasks.
When dealing with variables raised to exponents, such as \( y^5 \), this means \( y \) is multiplied by itself 5 times. When simplified within square roots, exponents need extra care. We used properties of exponents like \( (a^{m})^n = a^{m\cdot n} \) to break down and simplify complex expressions within roots. So, \( \sqrt{y^5} \) simplified becomes \( y^{2.5} \), using expression \( (y^{2.5})^2 = y^5 \).
Understanding how to manipulate exponents is key in simplification tasks.
Real Numbers
Real numbers include all the numbers on the number line. This means all integers, fractions, decimals, and irrational numbers. In algebra, especially when simplifying expressions, it's important to work with positive real numbers when dealing with square roots and variables.
This is because the square root of a negative number is not defined in the set of real numbers. When dealing with expressions such as \( 9y^5 \) and \( 25y^5 \), we assume \( y \) is a positive real number, as this ensures the expressions are solvable without leading into complex numbers.
Therefore, when the original exercise referred to all variables representing positive real numbers, it established the context needed for valid operations and simplifications.
This is because the square root of a negative number is not defined in the set of real numbers. When dealing with expressions such as \( 9y^5 \) and \( 25y^5 \), we assume \( y \) is a positive real number, as this ensures the expressions are solvable without leading into complex numbers.
Therefore, when the original exercise referred to all variables representing positive real numbers, it established the context needed for valid operations and simplifications.
Like Terms
Like terms are terms in algebra that have the same variable raised to the same power. Recognizing and combining like terms is a critical skill for simplifying expressions.
In the given exercise, once the square roots are simplified, we end up with terms like \( y^{2.5}, 3y^{2.5}, \) and \( 5y^{2.5} \). These are all like terms because each term contains \( y^{2.5} \).
This process simplifies the expression, making it easier to handle in further calculations or applications.
In the given exercise, once the square roots are simplified, we end up with terms like \( y^{2.5}, 3y^{2.5}, \) and \( 5y^{2.5} \). These are all like terms because each term contains \( y^{2.5} \).
- \( y^{2.5} \)
- \( 3y^{2.5} \)
- \( 5y^{2.5} \)
This process simplifies the expression, making it easier to handle in further calculations or applications.
Other exercises in this chapter
Problem 78
Simplify each expression. Write the answers without negative exponents. All variables represent positive real numbers. See Example 8. $$ 4^{2 / 5} \cdot 4^{2 /
View solution Problem 78
Divide. Write all answers in the form a \(+b i.\) $$ \frac{5-3 i}{4+2 i} $$
View solution Problem 79
Simplify each cube root. See Example 6. $$ \sqrt[3]{-216 a^{3}} $$
View solution Problem 79
Find the distance between each pair of points. $$ (\sqrt{48}, \sqrt{150}) \text { and }(\sqrt{12}, \sqrt{24}) $$
View solution