Question

part 1: please circle the answer and fill in the answer sheet on the last page. each question is worth 2 pts. 1. the expression $x^{2a + b}$ is equivalent to (1) $x^{2a}+x^{b}$ (2) $x^{a}+x^{a + b}$ (3) $x^{a}cdot x^{a + b}$ (4) $x^{a + b}cdot x^{a + b}$ 2. the expression is equivalent to $(5x^{2}-2x + 4)-(3x^{2}+3x - 1)$? 3. which expression results in a rational number? (1) $sqrt{3}cdotsqrt{2}$ (2) $-\frac{2}{3}+pi$ (3) $3cdotsqrt{10}$ (4) $\frac{1}{2}+sqrt{9}$ 4. the expression is equivalent to $4^{6}cdot4^{2}$? 5. when $6x^{3}-2x - 2$ is subtracted from $5x^{3}+3x - 4$, the result is 6. if the expression $(2y^{a})^{4}$ is equivalent to $16y^{16}$, what is the value of a? 7. the expression $\frac{-24x}{4x^{6}},x
eq0$, is equivalent to 8. what is the constant term of the polynomial $2 + 2x^{3}-x + 4x^{2}$?

Explanation:

Step1: Use exponent - rule \(x^{m + n}=x^{m}\cdot x^{n}\)

For \(x^{2a + b}=x^{a+(a + b)}=x^{a}\cdot x^{a + b}\), so the answer to question 1 is (3).

Step2: Expand the subtraction of polynomials

\((5x^{2}-2x + 4)-(3x^{2}+3x - 1)=5x^{2}-2x + 4-3x^{2}-3x + 1=(5x^{2}-3x^{2})+(-2x-3x)+(4 + 1)=2x^{2}-5x + 5\), so the answer to question 2 is \(2x^{2}-5x + 5\).

Step3: Recall the definition of rational numbers

\(\sqrt{3}\cdot\sqrt{2}=\sqrt{6}\), \(-\frac{2}{3}+\pi\) is irrational since \(\pi\) is irrational, \(3\cdot\sqrt{10}\) is irrational, \(\frac{1}{2}+\sqrt{9}=\frac{1}{2}+3=\frac{1 + 6}{2}=\frac{7}{2}\) is rational. So the answer to question 3 is (4).

Step4: Use the rule \(a^{m}\cdot a^{n}=a^{m + n}\)

\(4^{6}\cdot4^{2}=4^{6 + 2}=4^{8}\), so the answer to question 4 is \(4^{8}\).

Step5: Subtract the polynomials

\((5x^{3}+3x - 4)-(6x^{3}-2x - 2)=5x^{3}+3x - 4-6x^{3}+2x + 2=(5x^{3}-6x^{3})+(3x + 2x)+(-4 + 2)=-x^{3}+5x - 2\), so the answer to question 5 is \(-x^{3}+5x - 2\).

Step6: Expand \((2y^{a})^{4}\)

\((2y^{a})^{4}=2^{4}\cdot y^{4a}=16y^{4a}\), since \(16y^{4a}=16y^{16}\), then \(4a = 16\), \(a = 4\), so the answer to question 6 is \(a = 4\).

Step7: Simplify the rational - expression

\(\frac{-24x}{4x^{6}}=\frac{-24}{4}\cdot\frac{x}{x^{6}}=-6x^{1 - 6}=-6x^{-5}=-\frac{6}{x^{5}}\), so the answer to question 7 is \(-\frac{6}{x^{5}}\).

Step8: Identify the constant term

In the polynomial \(2+2x^{3}-x + 4x^{2}\), the constant term is the term without a variable, which is 2. So the answer to question 8 is 2.

Answer:

1. (3) \(x^{a}\cdot x^{a + b}\)
2. \(2x^{2}-5x + 5\)
3. (4) \(\frac{1}{2}+\sqrt{9}\)
4. \(4^{8}\)
5. \(-x^{3}+5x - 2\)
6. \(a = 4\)
7. \(-\frac{6}{x^{5}}\)
8. 2