Day: May 15, 2023

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202305151128 Exercise 1.1

Evaluate \textrm{\scriptsize{WITHOUT}} a calculator.

1. (-1)^4 2. (-1)^5 3. (-1)^{10} 4. (-1)^{15} 5. (-1)^8 6. -1^8 7. -(-1)^8 8. (-3)^3 9. -3^3 10. -(-3)^3 11. -(-6)^2 12. -(-4)^3 13. 2^3\times 3^2\times (-1)^5 14. (-1)^4\times 3^3\times 2^2 15. (-2)^3\times (-3)^4 16. 3^0 17. 6^{-1} 18. 4^{-1} 19. 5^0 20. 3^2 21. 3^{-2} 22. 5^3 23. 5^{-3} 24. 7^2 25. 7^{-2} 26. 10^3 27. 10^{-3}

Extracted from Phu Nielson. (2015). SAT Math Advanced Guide and Workbook.

Roughwork.

1.

Since (x^m)^n=x^{mn},

\begin{aligned} (-1)^4 & = ((-1)^2)^2 \\ & = (1)^2\\ & = 1 \\ \end{aligned}

2.

Since x^m\cdot x^n=x^{m+n},

\begin{aligned} (-1)^5 & = (-1)^{2(2)+1} \\ & = (-1)^{2(2)}\cdot (-1)^1 \\ & = ((-1)^{2})^2\cdot (-1)\\ & = (1)^2\cdot (-1) \\ & = 1\cdot (-1) \\ & = -1 \\ \end{aligned}

3.

\begin{aligned} (-1)^{10} & = (-1)^{(2)(5)}\\ & = ((-1)^2)^5 \\ & = (1)^5 \\ & = 1\\ \end{aligned}

4.

\begin{aligned} (-1)^{15} & = (-1)^{2(7)+1} \\ & = (-1)^{2(7)}\cdot (-1)^1 \\ & = ((-1)^2)^7\cdot (-1) \\ & = (1)^7\cdot (-1) \\ & = 1\cdot (-1) \\ & = -1 \\ \end{aligned}

5.

\begin{aligned} (-1)^8 & = (-1)^{2(4)} \\ & = ((-1)^2)^4 \\ & = (1)^4 \\ & = 1 \\ \end{aligned}

6.

\begin{aligned} -1^8 & = -(1^8) \\ & = -(1) \\ & = -1 \\ \end{aligned}

7.

\begin{aligned} -(-1)^8 & = -((-1)^8) \\ & = -((-1)^{2(4)}) \\ & = -(((-1)^2)^4) \\ & = -(1^4) \\ & = -(1) \\ & = -1 \\ \end{aligned}

8.

Since (xy)^m=x^my^m,

\begin{aligned} (-3)^3 & = ((-1)(3))^3 \\ & = (-1)^3(3)^3 \\ & = (-1)^{2+1}(3\cdot 3\cdot 3) \\ & = ((-1)^2\cdot (-1))(9\cdot 3) \\ & = (1\cdot (-1))(27) \\ & = (-1)(27) \\ & = -27 \\ \end{aligned}

9.

\begin{aligned} -3^3 & = -(3^3) \\ & = -(3\cdot 3\cdot 3) \\ & = -(9\cdot 3) \\ & = -(27) \\ & = -27 \\ \end{aligned}

10.

\begin{aligned} -(-3)^3 & = -((-3)^3) \\ & = -(((-1)(3))^3) \\ & = -((-1)^3(3)^3) \\ & = -((-1)^{2+1}(3\cdot 3\cdot 3)) \\ & = -((-1)^2(-1)(9\cdot 3)) \\ & = -((1)(-1)(27)) \\ & = -((-1)(27)) \\ & = -(-27) \\ & = 27 \\ \end{aligned}

Expressions 11. to 27. are not to be attempted.