Problem 11 - Algebra 1: Concepts and Skills

Question

Simplify the expression. \(5(-t)(-t)(-t)(-t)\)

Step-by-Step Solution

Answer

The simplified expression is \(5t^4 \)

Step 1: Understand the exercise

The expression we have is an algebraic multiplication \(5(-t)(-t)(-t)(-t)\). The aim is to simplify this expression.

Step 2: Distribute multiplication

Multiplication is associative, which means the way in which factors are grouped does not change the product. Therefore, the expression can be rewritten as follows: \(5 \times (-t) \times (-t) \times (-t) \times (-t)\)

Step 3: Multiply the negative terms

The product of two negative numbers is a positive number. Consequently, we can simplify \( (-t) \times (-t) \) to \( t^2 \). Now our expression becomes \(5 \times t^2 \times (-t) \times (-t)\).

Step 4: Continue multiplying the negative terms

Applying the same logic, we can further simplify \( t^2 \times (-t) \times (-t) \) to \( t^4 \). Our expression now simplifies to \(5 \times t^4 \)

Step 5: Final simplification

The expression \(5 \times t^4 \) can be rewritten to the simplified form \(5t^4 \)

Key Concepts

Algebraic Multiplication

When you see an expression like \(5(-t)(-t)(-t)(-t)\), you're dealing with algebraic multiplication. Algebraic multiplication involves combining numbers and variables by multiplying them. To simplify such expressions, you distribute the multiplication throughout each term. This means you multiply the numerical coefficients, like the 5 in this case, with the other factors systematically.
Remember that multiplication is associative, which means the order in which you multiply terms does not change the result. This property makes it possible to group terms in any order. Therefore, \(5(-t)(-t)(-t)(-t)\) is the same as \(5 \times (-t) \times (-t) \times (-t) \times (-t)\). By managing multiplication carefully, you ensure each step leads to a progressively simpler expression.

• Identify all the terms you need to multiply.
• Use associative property to regroup terms if necessary.
• Multiply each term accurately, ensuring every factor is considered.
• Always recheck for any additional simplifications possible.

Negative Numbers

When working with negative numbers in algebraic multiplication, it's important to remember basic rules that determine the sign of your result. Negative numbers, like \(-t\), may initially look intimidating, but they follow simple patterns:

• Multiplying two negative numbers yields a positive product. For example, \((-t)\times(-t) = t^2\).
• Multiplying a positive number and a negative number results in a negative product.

In this exercise, you multiply several negative numbers together, \((-t)(-t)(-t)(-t)\). Notice that multiplying each pair of negative numbers, the negatives cancel out:

• First pair: \((-t)\times(-t) = t^2\)
• Second pair: \((-t)\times(-t) = t^2\)

You apply these rules to systematically simplify until you combine all similar terms, ensuring you deal with the signs correctly and simplify the product to a simpler form \(5t^4\). Understanding these rules will help streamline your computations with negative expressions.

Exponents

Exponents are a shorthand way to express repeated multiplication of the same term. For example, \(t^4\) tells us that the variable \(t\) is multiplied by itself four times. In the given exercise \(5(-t)(-t)(-t)(-t)\), after identifying and simplifying products of negative terms, you use exponents to represent them neatly.
You start by computing \((-t)\times(-t) = t^2\), indicating \(t\) is multiplied by itself two times for now. Then, you multiply this by another \(t^2\) from further negative pairs, leading to \(t^2\times t^2 = t^4\).
Using the properties of exponents:

• When you multiply powers with the same base, you add their exponents. For example, \(t^a \times t^b = t^{a+b}\).

When you finish simplifying, you'll have \(5 \times t^4\), but this is often written more succinctly as \(5t^4\). This concise use of exponents simplifies algebraic manipulation, making expressions shorter and easier to work with.