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Chapter 4: Problem 61

Write each expression using one exponent. $$-4 \cdot t \cdot t \cdot t \cdot t$$

Short Answer

Expert verified
The simplified form of the given expression is \(-4 \cdot t^{4}\).

Step by step solution

01

Identify the base and counts of repetition

Looking at the expression \(-4 \cdot t \cdot t \cdot t \cdot t\), it can be seen that 't' is the base that's getting repeated and it's repeated 4 times.
02

Apply exponentiation rule

According to the rule of exponents, \(t \cdot t \cdot t \cdot t\) equals to \(t^{4}\).
03

Apply negative sign and constant term

Multiply the result from step 2 (\(t^{4}\)) with the constant term -4. This results in \(-4 \cdot t^{4}\).

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Key Concepts

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

Understanding Algebraic Expressions
Algebraic expressions are combinations of variables, constants, and arithmetic operations. In our exercise, we have an expression
  • Variable: Represented by 't', which is being multiplied multiple times.
  • Constant: The number -4 acts as a constant multiplier.
Recognizing these components is crucial. The variable part, here 't', can vary its value, whereas constants like -4 do not change. This combination of constants and variables, connected through multiplication and other operations, forms the essence of algebraic expressions. Understanding these will help in manipulating and simplifying expressions systematically.
The Base and Exponent
The terms 'base' and 'exponent' are integral in algebra, particularly when simplifying expressions using exponents.

In any term such as \[ t^4 \],
  • 't' is the base - it is the variable repeatedly multiplied.
  • '4' is the exponent - it tells us how many times to multiply the base by itself.
In the given exercise, 't' is multiplied by itself four times, therefore the expression of these repeated multiplications becomes \[ t^4 \].

Recognizing the base and exponent helps in efficiently expressing repetitive multiplication much more succinctly using exponential notation.
Dealing with Negative Constants
Negative constants can often change the sign and the magnitude of our results. In such expressions, like \[ -4 imes t^4 \],
  • The constant -4 signifies that everything it multiplies will end up being negative.
We begin by evaluating the expression \[ t imes t imes t imes t \], which simplifies to \[ t^4 \].

We then multiply the result by -4. Remember that multiplying by a negative constant reverses the sign, giving \[ -4 imes t^4 \].

This understanding is essential in maintaining the correct sign in expression manipulation.
Simplifying with Multiplication
Multiplication in algebra allows us to condense terms for simplicity. By using multiplication along with exponents, repetitive tasks are made efficient.

In the expression \[ -4 imes t imes t imes t imes t \],
  • We start by multiplying the variable 't' by itself \[ t imes t imes t imes t = t^4 \].
  • Next, we introduce the multiplication by -4.
By doing this, the entire expression can be simplified into a more compact form \[ -4 imes t^4 \].

Using multiplication allows us not only to simplify but also to clearly present the relationship between base, exponent, and constant.

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Most popular questions from this chapter

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