Find The Product. Simplify Your Answer.$(4g^4)(-2g$\] $\square$

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Understanding the Problem

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When it comes to simplifying expressions, one of the most common operations is finding the product of two or more terms. In this case, we are given the expression $(4g^4)(-2g)$ and we need to simplify it.

What is a Product in Algebra?

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In algebra, a product is the result of multiplying two or more terms together. When we multiply two terms, we are essentially combining their values to get a new value. For example, if we multiply 2 and 3, we get 6.

Simplifying the Expression

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To simplify the expression $(4g^4)(-2g)$, we need to multiply the two terms together. When we multiply two terms, we multiply their coefficients (the numbers in front of the variables) and their variables (the letters with exponents).

Multiplying Coefficients

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The coefficient of the first term is 4 and the coefficient of the second term is -2. When we multiply these two coefficients together, we get:

4 × -2 = -8

Multiplying Variables

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The variable of the first term is $g^4$ and the variable of the second term is $g$. When we multiply these two variables together, we need to add their exponents. This is because when we multiply two variables with the same base (in this case, g), we add their exponents.

$g^4$ × $g$ = $g^{4+1}$ = $g^5$

Combining the Results

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Now that we have multiplied the coefficients and variables together, we can combine the results to get the simplified expression:

-8 × $g^5$ = -8$g^5$

Conclusion

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In conclusion, to simplify the expression $(4g^4)(-2g)$, we need to multiply the two terms together. We multiply the coefficients and variables separately and then combine the results to get the simplified expression.

Tips and Tricks

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• When multiplying two terms, make sure to multiply the coefficients and variables separately.
• When multiplying variables with the same base, add their exponents.
• When combining the results, make sure to multiply the coefficients and variables together.

Common Mistakes

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• Forgetting to multiply the coefficients and variables separately.
• Not adding the exponents when multiplying variables with the same base.
• Not combining the results correctly.

Practice Problems

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• Simplify the expression $(3x^2)(-4x)$.
• Simplify the expression $(2y^3)(-5y)$.
• Simplify the expression $(4z^4)(-3z)$.

Final Thoughts

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Simplifying expressions is an important skill in algebra. By following the steps outlined in this article, you can simplify expressions like $(4g^4)(-2g)$ with ease. Remember to multiply the coefficients and variables separately, add the exponents when multiplying variables with the same base, and combine the results correctly. With practice, you will become more confident and proficient in simplifying expressions.

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Frequently Asked Questions

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In this article, we will answer some of the most common questions related to simplifying expressions, specifically the product of two or more terms.

Q: What is the difference between multiplying and adding in algebra?

A: In algebra, multiplying two terms means combining their values to get a new value, whereas adding two terms means combining their values to get a new value that is the sum of the two original values.

Q: How do I multiply two variables with the same base?

A: When multiplying two variables with the same base, you add their exponents. For example, $g^4$ × $g$ = $g^{4+1}$ = $g^5$.

Q: What is the rule for multiplying coefficients?

A: When multiplying two coefficients, you multiply them together. For example, 4 × -2 = -8.

Q: Can I simplify an expression by combining like terms?

A: Yes, you can simplify an expression by combining like terms. Like terms are terms that have the same variable and exponent. For example, $2x^2$ + $3x^2$ = $5x^2$.

Q: How do I know if an expression can be simplified?

A: An expression can be simplified if it contains like terms that can be combined. For example, $2x^2$ + $3x^2$ can be simplified, but $2x^2$ + $3y^2$ cannot be simplified.

Q: What is the difference between a product and a sum in algebra?

A: In algebra, a product is the result of multiplying two or more terms together, whereas a sum is the result of adding two or more terms together.

Q: Can I simplify an expression by canceling out common factors?

A: Yes, you can simplify an expression by canceling out common factors. For example, $2x^2$ ÷ $2x$ = $x$.

Q: How do I know if an expression can be simplified by canceling out common factors?

A: An expression can be simplified by canceling out common factors if it contains common factors that can be canceled out. For example, $2x^2$ ÷ $2x$ can be simplified, but $2x^2$ ÷ $3x$ cannot be simplified.

Q: What is the final step in simplifying an expression?

A: The final step in simplifying an expression is to combine the results of multiplying and adding, and to cancel out any common factors.

Q: Can I simplify an expression that contains fractions?

A: Yes, you can simplify an expression that contains fractions. To simplify a fraction, you can multiply the numerator and denominator by the same value to eliminate the fraction.

Q: How do I simplify an expression that contains exponents?

A: To simplify an expression that contains exponents, you can use the rules of exponents, such as $a^m$ × $a^n$ = $a^{m+n}$.

Q: Can I simplify an expression that contains radicals?

A: Yes, you can simplify an expression that contains radicals. To simplify a radical, you can multiply the radicand by the same value to eliminate the radical.

Q: What is the most important thing to remember when simplifying expressions?

A: The most important thing to remember when simplifying expressions is to follow the order of operations (PEMDAS) and to combine like terms.

Q: Can I simplify an expression that contains absolute values?

A: Yes, you can simplify an expression that contains absolute values. To simplify an absolute value, you can remove the absolute value sign and simplify the expression inside.

Q: How do I know if an expression can be simplified?

A: An expression can be simplified if it contains like terms that can be combined, or if it contains common factors that can be canceled out.

Q: What is the final answer to the expression $(4g^4)(-2g)$?

A: The final answer to the expression $(4g^4)(-2g)$ is -8$g^5$.

Q: Can I simplify an expression that contains parentheses?

A: Yes, you can simplify an expression that contains parentheses. To simplify an expression with parentheses, you can multiply the terms inside the parentheses and then simplify the expression.

Q: How do I simplify an expression that contains a negative exponent?

A: To simplify an expression that contains a negative exponent, you can rewrite the expression with a positive exponent by moving the base to the other side of the fraction.

Q: Can I simplify an expression that contains a zero exponent?

A: Yes, you can simplify an expression that contains a zero exponent. A zero exponent means that the base is equal to 1.

Q: What is the most common mistake when simplifying expressions?

A: The most common mistake when simplifying expressions is forgetting to multiply the coefficients and variables separately, or not adding the exponents when multiplying variables with the same base.

Q: Can I simplify an expression that contains a variable with a negative exponent?

A: Yes, you can simplify an expression that contains a variable with a negative exponent. To simplify an expression with a negative exponent, you can rewrite the expression with a positive exponent by moving the base to the other side of the fraction.

Q: How do I know if an expression can be simplified by canceling out common factors?

A: An expression can be simplified by canceling out common factors if it contains common factors that can be canceled out.

Q: What is the final step in simplifying an expression?

A: The final step in simplifying an expression is to combine the results of multiplying and adding, and to cancel out any common factors.

Q: Can I simplify an expression that contains a fraction with a variable in the denominator?

A: Yes, you can simplify an expression that contains a fraction with a variable in the denominator. To simplify a fraction with a variable in the denominator, you can multiply the numerator and denominator by the same value to eliminate the fraction.

Q: How do I simplify an expression that contains a variable with a fractional exponent?

A: To simplify an expression that contains a variable with a fractional exponent, you can use the rules of exponents, such as $a^{m/n}$ = $(a^m)^{1/n}$.

Q: Can I simplify an expression that contains a variable with a negative fractional exponent?

A: Yes, you can simplify an expression that contains a variable with a negative fractional exponent. To simplify an expression with a negative fractional exponent, you can rewrite the expression with a positive fractional exponent by moving the base to the other side of the fraction.

Q: What is the most important thing to remember when simplifying expressions?

A: The most important thing to remember when simplifying expressions is to follow the order of operations (PEMDAS) and to combine like terms.

Q: Can I simplify an expression that contains a variable with a complex exponent?

A: Yes, you can simplify an expression that contains a variable with a complex exponent. To simplify an expression with a complex exponent, you can use the rules of exponents, such as $a^{m+n}$ = $a^m$ × $a^n$.

Q: How do I simplify an expression that contains a variable with a negative complex exponent?

A: To simplify an expression that contains a variable with a negative complex exponent, you can rewrite the expression with a positive complex exponent by moving the base to the other side of the fraction.

Q: Can I simplify an expression that contains a variable with a complex fractional exponent?

A: Yes, you can simplify an expression that contains a variable with a complex fractional exponent. To simplify an expression with a complex fractional exponent, you can use the rules of exponents, such as $a^{m/n}$ = $(a^m)^{1/n}$.

Q: How do I simplify an expression that contains a variable with a negative complex fractional exponent?

A: To simplify an expression that contains a variable with a negative complex fractional exponent, you can rewrite the expression with a positive complex fractional exponent by moving the base to the other side of the fraction.

Q: What is the final answer to the expression $(4g^4)(-2g)$?

A: The final answer to the expression $(4g^4)(-2g)$ is -8$g^5$.