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Chapter 6: Problem 54

For Exercises \(49-54,\) divide. $$\frac{-2.7 t^{4} u^{3} v^{2}}{-0.75 u v^{2}}$$

Short Answer

Expert verified
Answer: The simplified expression is $3.6t^{4}u^{2}$.

Step by step solution

01

Write the division as multiplication with the divisor's reciprocal

To divide the given expression, rewrite the problem as a multiplication with the reciprocal of the divisor: $$\frac{-2.7 t^{4} u^{3} v^{2}}{-0.75 uv^{2}} \cdot \frac{1}{1} = \frac{-2.7 t^{4} u^{3} v^{2}}{1} \cdot \frac{1}{-0.75 uv^{2}}$$
02

Multiply the fractions

To perform the multiplication, multiply the numerators together and the denominators together: $$\frac{-2.7 t^{4} u^{3} v^{2}}{1} \cdot \frac{1}{-0.75 uv^{2}} = \frac{-2.7 t^{4} u^{3} v^{2} \cdot 1}{1 \cdot -0.75 uv^{2}}$$
03

Simplify the expression

Combine the coefficients and simplify the variable components: $$\frac{-2.7 t^{4} u^{3} v^{2}}{-0.75 uv^{2}} = \frac{(-2.7t^{4}u^{3}v^{2})(1)}{(-0.75uv^{2})(1)} = \frac{3.6t^{4}u^{2}}{1}$$ So, the simplified expression is: $$\frac{-2.7 t^{4} u^{3} v^{2}}{-0.75 u v^{2}}=3.6t^{4}u^{2}$$

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Key Concepts

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

Simplification
Simplification is a method used to reduce complex expressions into their simplest form.
This is achieved by canceling out common factors and combining like terms. In our polynomial expression, the goal was to simplify the division of two polynomials. By rewriting the problem into its simplest form, we managed to cancel out similar variables and factors in the numerator and the denominator.
To simplify any algebraic expression effectively, follow these steps:
  • Look for and cancel out common factors in both the numerator and the denominator.
  • Reduce coefficients by dividing them by their greatest common divisor.
  • Combine like terms where possible.
By understanding simplification, you can make solving polynomial problems much more manageable.
Fraction Multiplication
Fraction multiplication is fundamental in algebra, especially when dealing with division problems that involve fractions or polynomials.
This process is straightforward, as it involves multiplying the numerators together for a new numerator and the denominators together for a new denominator. For example, in our division problem, we converted division into multiplication by the reciprocal.
Here鈥檚 the breakdown:
  • First, identify the fractions involved in the division problem.
  • Replace the division sign with a multiplication sign.
  • Find the reciprocal of the second fraction before performing multiplication.
  • Multiply the numerators and then the denominators.
This method helps simplify the process and solve algebraic expressions more efficiently.
Reciprocal in Mathematics
A reciprocal in mathematics is a value that, when multiplied by the original number, results in one.
It is commonly used to simplify division problems, turning them into multiplication problems. For instance, the reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \).
To find a reciprocal:
  • Invert the fraction, exchanging the numerator and the denominator.
  • For whole numbers, write them as fractions by placing them over 1 and then invert.
Understanding the concept of reciprocals allows for an easy transition from division to multiplication, as seen in our initial exercise. It is especially useful when dealing with polynomial fractions, simplifying solutions significantly.

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Most popular questions from this chapter

Add or subtract. $$\left(6 a^{4}-3.1 a^{2}+1.2 a-1\right)-\left(5.4 a^{4}+1.7 a^{2}-2.9 a-1.32\right)$$

Add or subtract. $$\left(12.91 x^{3}-16.2 x^{2}-x+7\right)-\left(10 x^{3}-5.1 x^{2}-3.6 x+11.45\right)$$

A grain of salt is a cube that is 0.03 inch along each side. Calculate the surface area of a grain of salt.

Add or subtract. $$\left(5 k^{3}+2.5 k^{2}-6.2 k-0.44\right)-\left(2.2 k^{3}-6.2 k-0.5\right)$$

Solve and check. $$-16.7=h-7.59$$
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