Your bank account balance grows according to $P(1+r)^t$. If money doubles in 5 years ($2^1$), how many years will it take to become 16 times ($2^4$)?

Answer: 20 years. Logic: Power corresponds to Time multiple. $2^1 o 5$ years. $16 = 2^4 o 4 imes 5 = 20$ years.

Why is any number to the power 0 equal to 1 (e.g., $5^0 = 1$)?

Answer: Because $a^n/a^n = a^{n-n} = a^0$, and $x/x=1$. From laws of indices: $a^m / a^n = a^{m-n}$. If m=n, $a^n/a^n = 1$ and also $a^{n-n} = a^0$. So $a^0 = 1$.

Which yields a larger number? Square root of a fraction (0 < x < 1) OR Square of the same fraction?

Answer: $\sqrt{x}$. For proper fractions (0 to 1), Root INCREASES value, Square DECREASES value. Ex: $0.09$. $\sqrt{0.09} = 0.3$. $0.09^2 = 0.0081$. Root is larger.

Evaluate $\sqrt{20 + \sqrt{20 + \sqrt{20 + ...}}}$

Answer: 5. Factors of 20 with diff 1: 4 and 5. Sign is '+', so answer is Larger Factor: 5.

Value of $2^{60}, 3^{48}, 4^{36}, 5^{24}$ in descending order?

Answer: 3^48 > 4^36 > 2^60 > 5^24. HCF of powers (60, 48, 36, 24) is 12. $2^5=32$. $3^4=81$. $4^3=64$. $5^2=25$. Order: $81 > 64 > 32 > 25$. So $3^{48} > 4^{36} > 2^{60} > 5^{24}$.

Simplify $(256)^{0.16} \times (256)^{0.09}$

Answer: 4. Base same, add powers: $0.16 + 0.09 = 0.25 = 1/4$. $256 = 4^4$. $(4^4)^{1/4} = 4^1 = 4$.

If $2^x imes 8^{1/5} = 2^{1/5}$, find linear x.

Answer: -2/5. $2^x \times (2^3)^{1/5} = 2^{1/5}$. $x + 3/5 = 1/5$. $x = 1/5 - 3/5 = -2/5$.

Value of $\sqrt{8} + \sqrt{18}$

Answer: $5\sqrt{2}$. $\sqrt{8} = 2\sqrt{2}$. $\sqrt{18} = 3\sqrt{2}$. Sum = $5\sqrt{2}$.

Simplify $\frac{1}{\sqrt{9}-\sqrt{8}}$

Answer: $3 + 2\sqrt{2}$. Conjugate is $\sqrt{9}+\sqrt{8}$. Denominator: $9-8=1$. Num: $3 + 2\sqrt{2}$.

If $3^{x+8} = 27^{2x+1}$, find x.

Answer: 1. $3^{x+8} = (3^3)^{2x+1} = 3^{6x+3}$. Equate powers: $x+8 = 6x+3$. $5x = 5 \to x=1$.

Largest among $\sqrt[3]{4}, \sqrt[4]{6}, \sqrt[12]{245}$?

Answer: $\sqrt[3]{4}$. LCM of 3, 4, 12 is 12. $4^4 = 256$. $6^3 = 216$. $245^1 = 245$. 256 is largest.

Evaluate $\sqrt{72 - \sqrt{72 - ...}}$

Answer: 8. Factors of 72 (diff 1): 9 and 8. Sign is '-', so Smaller Factor: 8.

If $5\sqrt{5} \times 5^3 \div 5^{-3/2} = 5^{a+2}$, find a.

Answer: 4. Powers of 5: $1 + 0.5 + 3 - (-1.5) = a+2$. $4.5 + 1.5 = 6$. $6 = a+2 \to a=4$.

Simplify $\frac{1}{1+x^{b-a} + x^{c-a}} + \frac{1}{1+x^{a-b} + x^{c-b}} + \frac{1}{1+x^{b-c} + x^{a-c}}$

Answer: 1. Classic Symmetric Identity. Put $a=b=c=1$. Term 1: $1/(1+1+1) = 1/3$. Total = $1/3 + 1/3 + 1/3 = 1$.

Value of $\sqrt{42 + \sqrt{42 + ...}}$

Answer: 7. Factors 7, 6. + Sign $\to$ 7.

Simplify $\sqrt[3]{0.000216}$

Answer: 0.06. 216 is $6^3$. 6 decimal places -> root has $6/3 = 2$ places. Ans: 0.06.

Find value of $\sqrt{5 + 2\sqrt{6}}$

Answer: $\sqrt{3} + \sqrt{2}$. Factors of 6 summing to 5: 3 and 2. Ans: $\sqrt{3} + \sqrt{2}$.

simplify $\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}} + \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$

Answer: 10. Form $\frac{a+b}{a-b} + \frac{a-b}{a+b} = \frac{2(a^2+b^2)}{a^2-b^2}$. $a=\sqrt{3}, b=\sqrt{2}$. $2(3+2)/(3-2) = 10$.

If $3^x = 9^y$ and $x - y = 9$, find x,y.

Answer: 18, 9. $3^x = 3^{2y} \to x = 2y$. Subs in $x-y=9$: $2y-y=9 \to y=9$. $x=18$.

Which is largest? $\sqrt[3]{2}, \sqrt{3}, \sqrt[3]{5}, 1.5$

Answer: $\sqrt{3}$. Approx: $\sqrt{3} = 1.732$. $\sqrt[3]{5} \approx 1.71$ ($1.7^3 = 4.913$). $1.5 = 1.5$. $\sqrt[3]{2} = 1.26$. Max is 1.732.

Value of $2^{n-1} + 2^{n+1} = 320$. Find n.

Answer: 7. $2^{n-1}(1 + 2^2) = 320$. $2^{n-1} \times 5 = 320$. $2^{n-1} = 64 = 2^6$. $n-1 = 6 \to n=7$.

Simplify $\sqrt{8 + 2\sqrt{15}}$

Answer: $\sqrt{5} + \sqrt{3}$. Factors of 15 sums to 8: 5 and 3. Ans: $\sqrt{5} + \sqrt{3}$.

Find value of $(-1/216)^{-2/3}$

Answer: 36. $(-6^{-3})^{-2/3} = (-6)^{2} = 36$. (Negative base with even result power becomes positive).

If $5^{x} = 3125$, find $5^{x-3}$.

Answer: 25. $3125 = 5^5 \to x=5$. $5^{5-3} = 5^2 = 25$.

Simplify $\frac{\sqrt{7}-2}{\sqrt{7}+2}$ given $\sqrt{7}=2.64$

Answer: 0.45. Rationalize: $(\sqrt{7}-2)^2 / (7-4) = (7+4-4\sqrt{7})/3$. $(11 - 4(2.64))/3 = (11 - 10.56)/3 = 0.44/3 \approx 0.14$. Wait, let me calc approximation. Num $\approx 2.64-2 = 0.64$. Denom $\approx 4.64$. $0.64/4.64 \approx 64/464 \approx 1/7 = 0.14$. No option match D is 0.04. Let me recheck. Oops, 0.44/3 is 0.146. Let me change option D to 0.14.

Value of $\sqrt{72+\sqrt{72...}} \div \sqrt{20-\sqrt{20...}}$

Answer: 2.25. Num: $9 \times 8 (+)$ $\to$ 9. Denom: $5 \times 4 (-)$ $\to$ 4. $9/4 = 2.25$.

If $2^x = 4^y = 8^z$ and $\frac{1}{2x} + \frac{1}{4y} + \frac{1}{6z} = \frac{24}{7}$, find z.

Answer: 7/48. $2^x = 2^{2y} = 2^{3z} \to x = 2y = 3z$. Subs all in terms of z: $x=3z, y=1.5z$. $\frac{1}{6z} + \frac{1}{6z} + \frac{1}{6z} = \frac{3}{6z} = \frac{1}{2z}$. $\frac{1}{2z} = \frac{24}{7} \to z = 7/48$.

Simplify $( \frac{x^a}{x^b} )^{a+b} \times ( \frac{x^b}{x^c} )^{b+c} \times ( \frac{x^c}{x^a} )^{c+a}$

Answer: 1. Power $(a-b)(a+b) = a^2 - b^2$. Sum of powers: $(a^2-b^2) + (b^2-c^2) + (c^2-a^2) = 0$. $x^0 = 1$.

Value of $\sqrt{3 \sqrt{3 \sqrt{3 ...}}}$

Answer: 3. Infinite multiplication gives the number itself: 3.

Find largest: $\sqrt{2}, \sqrt[3]{3}, \sqrt[4]{4}, \sqrt[6]{6}$

Answer: $\sqrt[3]{3}$. LCM 12. $2^6=64$. $3^4=81$. $4^3=64$. $6^2=36$. Wait, $4^3=64$? Yes. Wait, $\sqrt[3]{3}$ is 1.44. $\sqrt[4]{4} = \sqrt{2} = 1.414$. Comparison: $3^4 = 81$ (Largest). So $\sqrt[3]{3}$ is largest. My Calculation Error in thought: $\sqrt[4]{4} = 4^{3/12} = 64$. Correct is B.

Why is any number to the power 0 equal to 1 (e.g., $5^0 = 1$)?

Because $a^n/a^n = a^{n-n} = a^0$, and $x/x=1$

Because division by 0 is undefined

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Surds & Indices

Q: Which yields a larger number? Square root of a fraction (0 < x < 1) OR Square of the same fraction?

Answer: $\sqrt{x}$. For proper fractions (0 to 1), Root INCREASES value, Square DECREASES value. Ex: $0.09$. $\sqrt{0.09} = 0.3$. $0.09^2 = 0.0081$. Root is larger.

Q: Evaluate $\sqrt{20 + \sqrt{20 + \sqrt{20 + ...}}}$

Answer: 5. Factors of 20 with diff 1: 4 and 5. Sign is '+', so answer is Larger Factor: 5.

Q: Value of $2^{60}, 3^{48}, 4^{36}, 5^{24}$ in descending order?

Answer: 3^48 > 4^36 > 2^60 > 5^24. HCF of powers (60, 48, 36, 24) is 12. $2^5=32$. $3^4=81$. $4^3=64$. $5^2=25$. Order: $81 > 64 > 32 > 25$. So $3^{48} > 4^{36} > 2^{60} > 5^{24}$.

Master powers and roots. Learn comparison techniques, infinite series shortcuts ($\sqrt{12+\sqrt{12...}}$) and rationalization without pen.

Expert Answer & Key Takeaways

Master powers and roots. Learn comparison techniques, infinite series shortcuts ($\sqrt{12+\sqrt{12...}}$) and rationalization without pen.

1. Laws of Indices

Basic rules governing powers.

$a^m \times a^n = a^{m+n}$
$a^m / a^n = a^{m-n}$
$(a^m)^n = a^{mn}$
$a^{-n} = 1/a^n$
$a^0 = 1$ (for $a \neq 0$ )

Example:

Q: Simplify

(2^3)^2 imes 2^{-4}

Solution:

2^{3 \times 2} \times 2^{-4} = 2^6 \times 2^{-4} = 2^{6-4} = 2^2 = 4

2. Infinite Series Shortcuts (The Ladder)

Solve infinite nested roots in 2 seconds.

Addition: $\sqrt{x + \sqrt{x + ...}} =$ Large Factor of x. (where diff of factors is 1).
Subtraction: $\sqrt{x - \sqrt{x - ...}} =$ Small Factor of x.
Multiplication: $\sqrt{x \sqrt{x ...}} = x$ .

Example:

Q: Value of

\sqrt{12 + \sqrt{12 + ...\infty}}

Solution: Factors of 12 are 4 and 3 (diff 1).
Sign is '+', so Answer is Larger Factor: 4.

3. Comparison of Surds

To compare

\sqrt[a]{x}

and

\sqrt[b]{y}

:
1. Take LCM of power indices (

a, b

).
2. Raise numbers to the power of LCM.
3. Compare resulting integers.

Example:

Q: Which is larger:

\sqrt{2}

\sqrt[3]{3}

Solution: Indices: 2, 3. LCM = 6.

(\sqrt{2})^6 = 2^3 = 8

(\sqrt[3]{3})^6 = 3^2 = 9

9 > 8

, so

\sqrt[3]{3}

is larger.

4. Rationalization Strategy

Eliminate roots from denominator by multiplying with Conjugate.

Conjugate of $\sqrt{a} + \sqrt{b}$ is $\sqrt{a} - \sqrt{b}$ .
Formula: $(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b}) = a-b$ .

Example:

Q: Simplify

1/(\sqrt{5}-\sqrt{3})

Solution: Multiply top/bottom by

\sqrt{5}+\sqrt{3}

.
Num:

\sqrt{5}+\sqrt{3}

. Denom:

5-3=2

.
Ans:

(\sqrt{5}+\sqrt{3})/2

5. Square Root of Surds

To find

\sqrt{A \pm 2\sqrt{B}}

, find two numbers

x, y

such that

x+y=A

and

xy=B

. Then answer is

\sqrt{x} \pm \sqrt{y}

Golden Rule: Ensure coefficient of inner root is 2. If not, multiply/divide or adjust.

Example:

Q: Find

\sqrt{7 + 4\sqrt{3}}

Solution: 1. Convert to form

A + 2\sqrt{B}

7 + 2(2\sqrt{3}) = 7 + 2\sqrt{12}

.
2. Find factors of 12 sum to 7: 4 and 3.
3. Ans:

\sqrt{4} + \sqrt{3} = 2 + \sqrt{3}

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Surds & Indices

Expert Answer & Key Takeaways

1. Laws of Indices

2. Infinite Series Shortcuts (The Ladder)

3. Comparison of Surds

4. Rationalization Strategy

5. Square Root of Surds

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