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Chapter 5: Problem 53

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

Short Answer

Expert verified
The answer is \(8.4y^7\).

Step by step solution

01

Distribute the Multiplication

To solve the expression \(-1.2y(-7y^6)\), you first distribute the multiplication by multiplying the coefficients and the variables separately. Start by recognizing that the expression consists of the multiplication of \(-1.2y\) with \(-7y^6\).
02

Multiply the Coefficients

Focus on the numerical coefficients: multiply \(-1.2\) and \(-7\). This results in \(8.4\) since \(-1.2\) times \(-7\) equals positive \(8.4\) (as the product of two negative numbers is positive).
03

Multiply the Variables

Now look at the variable part: multiply \(y\) by \(y^6\). Using the property of exponents, where \(a^m \times a^n = a^{m+n}\), the result is \(y^{1+6} = y^7\).
04

Combine Results

Combine the results of the coefficient multiplication and the variable multiplication. You get \(8.4y^7\) as the simplified form of the expression.

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Key Concepts

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

Algebraic Expressions
An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. In this particular exercise, the expression \(-1.2y(-7y^6)\) is examined.
The components of this expression include:
  • **Coefficients:** Numbers that are multiplied by the variables, such as \(-1.2\) and \(-7\).
  • **Variables:** Symbols like \(y\) which represent unknown values.
  • **Operations:** In this case, multiplication is the operation that combines the terms.
Understanding each part's role helps simplify or manipulate the expression easily. This process lays the groundwork for using more complex algebraic skills. Translating verbal statements into algebraic expressions is essential in solving real-world problems.
Exponent Rules
Exponents are shorthand for repeated multiplication of a number by itself. When working with algebraic expressions, exponent rules become crucial, especially during multiplication. In our exercise, \(y\) and \(y^6\) are multiplied together.
There are a few key exponent rules:
  • The Product of Powers Rule: \(a^m \times a^n = a^{m+n}\). For example, multiplying \(y\) (which is \(y^1\)) and \(y^6\) gives \(y^{1+6} = y^7\).
  • A Power of a Power: \(a^{m \times n}\) is \(a^{mn}\).
  • Zero Exponent Rule: Any number raised to the power of zero is 1, \(a^0 = 1\), except when \(a\) is zero.
Utilizing these rules streamlines multiplication and helps prevent errors, making it easier to understand larger expressions.
Coefficient Multiplication
Coefficient multiplication involves multiplying the numerical parts of algebraic terms. In the exercise, we multiply \(-1.2\) with \(-7\).
Key points to remember when multiplying coefficients are:
  • **Signs:** When multiplying two negative numbers, the result is positive, as evidenced by \(-1.2 \times -7 = 8.4\).
  • **Decimal Multiplication:** It's important to line up decimal points properly and handle them just as you would with whole numbers.
  • **Combining Coefficients and Variables:** After calculating the product, recombine with your variable part to form the solution, such as \8.4y^7\ in our problem.
Breaking multiplication into smaller steps to handle coefficients simplifies solving polynomial expressions and aids in keeping calculations accurate.

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Most popular questions from this chapter

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