Problem 122 - Elementary and Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. $$ t^{4 / 3}\left(t^{5 / 3}+t^{-4 / 3}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( t^3 + 1 \).

Step 1: Distribute the Exponent

Apply the distributive property by multiplying \( t^{4/3} \) with each term inside the parentheses. Start with \( t^{4/3} \cdot t^{5/3} \).

Step 2: Add Exponents for the First Term

When multiplying like bases, add their exponents: \( t^{4/3} \cdot t^{5/3} = t^{(4/3) + (5/3)} = t^{9/3} = t^{3} \).

Step 3: Distribute to the Second Term

Multiply \( t^{4/3} \) by \( t^{-4/3} \) using the property of exponents: \( t^{4/3} \cdot t^{-4/3} = t^{(4/3) + (-4/3)} = t^0 \).

Step 4: Simplify Using Zero Exponent Rule

Any nonzero number raised to the power of zero is 1, so \( t^0 = 1 \).

Step 5: Combine the Simplified Terms

After simplifying both terms, the expression becomes \( t^3 + 1 \).

Key Concepts

Properties of Exponents

The properties of exponents are handy rules allowing us to simplify expressions involving powers.
When multiplying two expressions that have the same base, we add their exponents.
For example, in the expression \( t^{4/3} \cdot t^{5/3} \), both terms have the base \( t \). We add the exponents together like this: \( \frac{4}{3} + \frac{5}{3} \).
This results in \( t^{9/3} \), which simplifies to \( t^3 \). Understanding this property makes simplifying expressions with exponents much easier.

• When multiplying like bases: add the exponents.
• When dividing like bases: subtract the exponents.
• Raising a power to a power: multiply the exponents.

Mastering these tricks will make algebra much more manageable!
Onward to the joys of the distributive property.

Distributive Property

The distributive property is a fundamental concept that helps simplify expressions and solve equations. It says that if you have a term outside parentheses, you can "distribute" it by multiplying it with each term inside the parentheses.
In math language, \[a(b + c) = ab + ac\]. In the original problem, we applied the distributive property by multiplying \( t^{4/3} \) with each term inside the parentheses:

• Firstly, with \( t^{5/3} \).
• Secondly, with \( t^{-4/3} \).

This allows us to simplify the expression efficiently by dealing with smaller pieces one step at a time.

The distributive property is not just a trick but a bridging concept that is applicable across many areas of math and algebra.

Zero Exponent Rule

The zero exponent rule is quite intuitive once you get the hang of it. This rule states that any nonzero number raised to the power of zero is equal to 1.
In our expression, after applying the distributive property, one of the results was \( t^0 \). According to the zero exponent rule, this simplifies straightforwardly to 1.
It may seem mysterious, but here's a quick reasoning: when we multiply two terms with the same base, we essentially add the exponents. If you have \( t^{x} \cdot t^{0} = t^{x+0} \), which simplifies to \( t^x \), implying that \( t^0 \) doesn't change any multiplication value, i.e., it equals 1. It's crucial to remember that this rule only applies to nonzero bases.

• \( t^0 = 1 \) if \( t eq 0 \)
• Zero raised to the power of zero is not typically defined in elementary algebra.

Understanding this rule helps make sense of how exponents work and why they simplify in certain ways.