Problem 59 - Algebra A Combined Function

Question

Multiply. \(-1.2 y\left(-7 y^{6}\right)\)

Step-by-Step Solution

Answer

8.4y^7

Step 1: Multiply the Coefficients

First, identify and multiply the numerical coefficients from each term in the expression. This gives:- The coefficient from the first term is \(-1.2\).- The coefficient from the second term is \(-7\).- Multiply these coefficients: \(-1.2 imes -7 = 8.4\).

Step 2: Multiply the Variables

Next, look at the variable part of the expression, which is \(y\) raised to some power:- From the first term, there is one \(y\).- From the second term, \(y\) is raised to the power of 6, or \(y^6\).- To multiply these, add the exponents: \(y^{1+6} = y^7\).

Step 3: Combine Results

Finally, combine the result from multiplying the coefficients with the result from multiplying the variables:- The coefficient result is \8.4\.- The variable result is \(y^7\).- Thus, the complete expression is \8.4y^7\.

Key Concepts

Coefficients Multiplication

To understand algebraic expressions multiplication, especially involving coefficients, we need to focus on the numerical parts of terms first. Coefficients are numbers that are directly multiplied by variables, influencing the magnitude of the term. In this exercise, we start with the coefficients: the number \(-1.2\) from the first term and \(-7\) from the second term.

Multiplying coefficients is straightforward. Negative signs in the coefficients can change the sign of the result:

• Here, \(-1.2 \times -7 = 8.4\). Note how multiplying two negatives results in a positive.
• This multiplication of coefficients simplifies the numerical part of the expression.

Understanding how to handle numbers, especially with different signs, is crucial. Keep track of any negative signs as they affect the entire expression.

Exponents Addition

When multiplying variables, especially with exponents, it's important to handle the powers correctly. In algebra, each variable may be raised to a power, known as an exponent.

In our exercise, we are dealing with the variable \('y'\):

• The first term is simply \('y'\) which is implicitly \(y^1\).
• The second term is \(y^6\).

To multiply these variables, we add their exponents:

• The rule for multiplying variables is to add their exponents together: \(y^{1+6} = y^7\).
• This addition comes from the law of exponents and helps in simplifying the expression.

This process of adding exponents is an essential step in simplifying expressions in algebra.
By mastering this, any algebraic multiplication problem becomes easier to solve.

Variables Multiplication

Multiplying variables involves a clear understanding of what the variables represent and how they work together. The primary variable in this expression is \('y'\). Every time we multiply similar variables, we need to keep their exponents in mind. When expressed as a term, these multiply together to form a single like variable.

Consider:

• The first term contains \('y'\), considered as \(y^1\).
• The second term contains \(y^6\).

When multiplying these, we add their exponents: \(y^{1+6} = y^7\).
This indicates how often the base variable, in this case \('y'\), is used as a factor in the expression.
Understanding the multiplication of variables in algebra not only enhances problem-solving skills but also solidifies comprehension of how algebraic terms become combined into simplified forms.

Combining Results in Algebra

Once the coefficients and the variable parts have been calculated, we need to combine the results to form a single, simplified algebraic expression. This final step in algebraic multiplication tidies up the entire equation.

Here's the summary of our work:

• We found the coefficient result to be \(8.4\).
• The variables, after applying exponents, resulted in \(y^7\).

Thus, the full expression is simplified into \(8.4y^7\).
Combining results after breaking the problem into manageable sections is common in algebra. It not only simplifies problem-solving but also ensures accuracy in computations.
This skill is essential for tackling more complex algebraic operations, laying a foundation for mastering sophisticated math problems later on.