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1.a Evaluate:
$\displaystyle \sqrt{1\frac{11}{25}}$
first change what is under the root symbol from a mixed number to an
improper fraction:
$\displaystyle
\sqrt{1\frac{11}{25}}=\sqrt{\frac{36}{25}}=\frac{ \sqrt {36}}{\sqrt{25}}
$
You should now be able to complete this without a calculator as
both $\displaystyle 36$ and $\displaystyle 25$ are perfect squares.
RonL
1.b Evaluate $\displaystyle \sqrt{2120}$ without a calculator given that
$\displaystyle \sqrt{2.12}=1.456$ and $\displaystyle \sqrt{21.2}=4.004$.
We want to write $\displaystyle 2120$ as a product of a square and one of
the number we are given the square root of. Now:
$\displaystyle 2120=1000 \times 2.12$,
and
$\displaystyle 2120=100 \times 21.2$.
$\displaystyle 1000$ is not the square of an integer, but $\displaystyle 100=10^2$. So we can write:
$\displaystyle
\sqrt{2120}=\sqrt{100 \times 21.2}=\sqrt{100}\times \sqrt{21.2}
$.
RonL
1.c Evaluate without a calculator:
$\displaystyle
\left(-\frac{1}{216}\right)^{-2/3}
$
Now:
$\displaystyle
\left(-\frac{1}{216}\right)^{-2/3}=\frac{1}{(-\frac{1}{216})^{2/3}}
$$\displaystyle
=\frac{1}{((-\frac{1}{216})^{1/3})^2}=\frac{1}{((-\frac{1}{(216)^{1/3}}))^2}
$
Now this can be evaluated from the innermost brackets outward, as 216 is
a perfect cube (of a small number which you can find by trial and error).
RonL
Hello,xiaoz!
Here's #6 . . .
We have: .$\displaystyle 5 \;< \; 6x - 13 \;< \;7$6(a) Solve the inequality: $\displaystyle 5 \;< \;6x - 13 \;< \;7$
Add 13 to each side: .$\displaystyle 18 \:< \:6x \:< \:20$
Divide by 6: . $\displaystyle 3 \;< \;x \;< \;\frac{10}{3}$
There are several ways to solve this problem.(b) Find a fraction $\displaystyle x$ such that: $\displaystyle \frac{7}{11}\;<\:x\:<\:\frac{8}{11}$
(1) To find a number between two given numbers, average the numbers.
. . . $\displaystyle x\;=\;\frac{\frac{7}{11} + \frac{8}{11}}{2} \;=\;\frac{15}{22}$
(2) Convert to decimals: $\displaystyle \begin{array}{cc}\frac{7}{11} = 0.63\hdots \\ \frac{8}{11} = 0.72\cdots\end{array}$
Then let $\displaystyle x$ be any decimal between $\displaystyle 0.64$ and $\displaystyle 0.72$
For example: . $\displaystyle x\:=\:0.65\:=\:\frac{65}{100} \:= \:\frac{13}{20}$
Change the first expression into the second, step by step.(c) If $\displaystyle x^{2y} = 5$, find the value of $\displaystyle 2x^{6y} - 4$
We have: .$\displaystyle x^{2y}\:=\:5$
Cube both sides: .$\displaystyle (x^{2y})^3\:=\:5^3\quad\Rightarrow\quad x^{6y}\:=\:125$
Multiply by 2: .$\displaystyle 2x^{6y}\;=\;250$
Subtract 4: .$\displaystyle 2x^{6y} - 4 \:= \:246$
Hello, xiaoz!
Let me give 1(c) a try . . .
Did you know that a negative exponent on a fraction flips the fraction?1(c) Simplify: $\displaystyle \left(-\frac{1}{216}\right)^{-\frac{2}{3}}$
. . $\displaystyle \left(\frac{a}{b}\right)^{-n}\;=\;\left(\frac{b}{a}\right)^n$
We have: $\displaystyle \left(-\frac{1}{216}\right)^{-\frac{2}{3}}\:=\: (-216)^{\frac{2}{3}} $
We know that: $\displaystyle -216 = (-6)^3$
So we have: $\displaystyle [(-6)^3]^{\frac{2}{3}}\;=\;6^{(3\cdot\frac{2}{3})} \;=\;6^2\;=\;36$