Problem 53
Question
\(47-72\) . Simplify the expression, and eliminate any negative exponent(s). $$ \left(s^{-2} t^{2}\right)^{2}\left(s^{2} t\right)^{3} $$
Step-by-Step Solution
Verified Answer
The simplified expression is \(s^2 t^7\).
1Step 1: Simplify the Exponents in Each Term
Start by distributing the exponents in each term using the power of a power property, which states that (\(a^m\))^n = \(a^{m \cdot n}\).\[(s^{-2} t^2)^2 = s^{-4} t^4\]\[(s^2 t)^3 = s^6 t^3\]
2Step 2: Combine the Simplified Terms
Multiply the results from Step 1 using the property of exponents that states \(a^m \cdot a^n = a^{m+n}\).\[s^{-4} t^4 \cdot s^6 t^3 = s^{(-4+6)} t^{(4+3)} = s^2 t^7\]
3Step 3: Final Expression Without Negative Exponents
The combined expression from Step 2 does not have negative exponents, so no further simplification is needed.The simplified expression is:\[s^2 t^7\]
Key Concepts
Power of a Power PropertyProperties of ExponentsEliminating Negative Exponents
Power of a Power Property
When you raise an exponent to another exponent, you use the power of a power property. This property is like a rule that helps you figure out these kinds of expressions easily. You might come across something like \((a^m)^n\).
To simplify this, you just multiply the exponents: \(a^{m \cdot n}\).
It’s a straightforward shortcut to avoid expanding everything manually.
In our exercise, we used the power of a power rule to simplify \((s^{-2} t^2)^2\).
That turned into \(s^{-4} t^4\) because for each variable inside the parentheses, we multiplied the exponents.
Similarly, \((s^2 t)^3\) became \(s^6 t^3\).
It’s like carrying a stack of numbers and, instead of adding a new layer each time, you just multiply the thickness all at once.
To simplify this, you just multiply the exponents: \(a^{m \cdot n}\).
It’s a straightforward shortcut to avoid expanding everything manually.
In our exercise, we used the power of a power rule to simplify \((s^{-2} t^2)^2\).
That turned into \(s^{-4} t^4\) because for each variable inside the parentheses, we multiplied the exponents.
Similarly, \((s^2 t)^3\) became \(s^6 t^3\).
It’s like carrying a stack of numbers and, instead of adding a new layer each time, you just multiply the thickness all at once.
Properties of Exponents
Exponents have several properties that can simplify expressions, making math a bit less daunting. Here are some of the crucial ones:
In our exercise, after simplifying each term with the power of a power property, we used the product of powers property.
We combined \(s^{-4} t^4\) with \(s^6 t^3\). This property helped us combine like terms:
Use these properties as tools, like little calculators, to quickly handle exponents.
- Product of Powers Property: When you multiply like bases, add the exponents.
For \(a^m \cdot a^n\), this becomes \(a^{m+n}\). - Power of a Power Property: Discussed above, this involves multiplying exponents when raising a power to a power.
In our exercise, after simplifying each term with the power of a power property, we used the product of powers property.
We combined \(s^{-4} t^4\) with \(s^6 t^3\). This property helped us combine like terms:
- For \(s\): \(s^{-4} \cdot s^6 = s^{(-4+6)} = s^2\)
- For \(t\): \(t^4 \cdot t^3 = t^{(4+3)} = t^7\)
Use these properties as tools, like little calculators, to quickly handle exponents.
Eliminating Negative Exponents
Exponents can sometimes be negative, and simplifying expressions usually means turning those negative exponents into positive ones.
A negative exponent indicates the reciprocal of the base. For example, \(a^{-n} = \frac{1}{a^n}\).
This is handy for making expressions look cleaner and more standard in mathematical solutions.
In our exercise walkthrough, we actually didn't end up with any negative exponents to remove!
But if we had to eliminate a negative exponent, we could have used the property to transform that part of the expression.
Often, you'll start with a mixed expression, apply the rules, and finally adjust any negative exponents to make everything positive.
Just flip that part of your expression to its reciprocal, and you’re all set!
A negative exponent indicates the reciprocal of the base. For example, \(a^{-n} = \frac{1}{a^n}\).
This is handy for making expressions look cleaner and more standard in mathematical solutions.
In our exercise walkthrough, we actually didn't end up with any negative exponents to remove!
But if we had to eliminate a negative exponent, we could have used the property to transform that part of the expression.
Often, you'll start with a mixed expression, apply the rules, and finally adjust any negative exponents to make everything positive.
Just flip that part of your expression to its reciprocal, and you’re all set!
Other exercises in this chapter
Problem 52
Multiply the algebraic expressions using a Special Product Formula, and simplify. \((1-2 y)^{2}\)
View solution Problem 53
Perform the addition or subtraction and simplify. $$ u+1+\frac{u}{u+1} $$
View solution Problem 53
Factor the expression completely. $$ t^{2}-6 t+9 $$
View solution Problem 53
Evaluate each expression. $$ \begin{array}{ll}{\text { (a) }|100|} & {\text { (b) }|-73|}\end{array} $$
View solution