Problem 84 - Algebra A Combined Function

Question

Simplify each expression. $$ \left(-3 s^{5} t\right)\left(-7 s t^{10}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(21 s^6 t^{11}\).

Step 1: Apply the Distributive Property

To begin simplifying, use the distributive property which involves multiplying each term from the first expression by each term in the second expression. Given the expression, multiply -3 with -7. The result is positive 21. So we have:\[21 (s^5 t) (s t^{10})\]

Step 2: Apply the Power Rule

Now, apply the power rule for exponents, which states that when you multiply like bases you add the exponents. First, handle the \(s\) terms: \[s^5 \times s = s^{5+1} = s^6\]Next, handle the \(t\) terms:\[t \times t^{10} = t^{1+10} = t^{11}\]Our expression now becomes:\[21 s^6 t^{11}\]

Step 3: Write the Final Simplified Expression

Once all like terms are combined by applying the power rule, the expression is fully simplified. Therefore, the fully simplified expression for the original problem is:\[21 s^6 t^{11}\]

Key Concepts

distributive property

The distributive property is a fundamental concept in algebra that helps to simplify expressions by distributing, or multiplying, each term inside a parenthesis by a factor outside the parenthesis. In the context of the exercise, the distributive property allows us to multiply the first expression, \(-3s^5t\), by the second expression, \(-7st^{10}\). This property essentially states that \(a(b+c) = ab + ac\).
In this example, we apply the distributive property like this:

• Multiply \(-3\) by \(-7\), resulting in \(21\).
• Multiply the variables \(s^5t\) and \(st^{10}\) using the power rule (explained later).

This step turns our expression into \[21(s^5t)(st^{10})\].
Remember, distributing means you treat each term in the parenthesis by itself and multiply it by the outside number or variable."

power rule

The power rule for exponents is a shortcut used to simplify expressions involving powers with the same base. When you multiply terms with the same base, you simply add their exponents. This rule is expressed as \(x^a \times x^b = x^{a+b}\).
In our example, the power rule is applied separately to the \(s\) and \(t\) terms:

• For \(s\), multiply \(s^5\) by \(s^1\), which gives \(s^{5+1} = s^6\).
• For \(t\), multiply \(t^1\) by \(t^{10}\), which results in \(t^{1+10} = t^{11}\).

The power rule simplifies complex expressions by reducing them to a single term for each variable, making calculations straightforward and manageable."

exponents

Exponents are a key mathematical concept that indicates how many times a number, known as the base, is multiplied by itself. In algebra, they help to compactly represent repeated multiplication. For instance, \(s^5\) means \(s\) is multiplied by itself five times.
Understanding how to manipulate exponents through multiplication or division is crucial. Here are some important rules:

• Multiplying like bases: Add their exponents \(x^a \times x^b = x^{a+b}\).
• Dividing like bases: Subtract their exponents \(x^a / x^b = x^{a-b}\).
• Power of a power: Multiply the exponents \((x^a)^b = x^{a \times b}\).

Exponents simplify the expression \(21s^6t^{11}\) by reducing the complex multiplication of terms to a single, manageable form."

like terms

Like terms are terms within an algebraic expression that share the same variables raised to the same power. When simplifying expressions, combining like terms ensures the expression is not overcomplicated. For example, \(5s^2t\) and \(3s^2t\) are like terms because they have the same variables \(s^2t\).
In the exercise, we combine the like terms using the power rule:

• Combine \(s^5\) with \(s\) as they have the common base, resulting in \(s^6\).
• Combine \(t\) with \(t^{10}\) to form \(t^{11}\).

Combining like terms reduces the complexity of expressions and ensures clarity, as it gives a single term that represents the total quantity of each distinct variable form in the expression."