United Kingdom arithmetic of exponents (negative exponents) Math - Free Printable

Let's solve each question step-by-step from the worksheet on Exponents and Powers for Class 8.

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(1) $ 3.8 \times 10^{11} + 3 \times 10^{12} = ? $

We need to add these two numbers in scientific notation.

First, make the exponents the same:

$$
3.8 \times 10^{11} + 3 \times 10^{12} = 3.8 \times 10^{11} + 30 \times 10^{11} = (3.8 + 30) \times 10^{11} = 33.8 \times 10^{11}
$$

Now convert to proper scientific notation:

$$
33.8 \times 10^{11} = 3.38 \times 10^{12}
$$

✔ Answer: $ 3.38 \times 10^{12} $
(Note: This is not listed among options, so perhaps there’s a typo or misprint. But based on calculation, this is correct.)

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(2) Write following numbers in usual form:

Options:
A) $ 4.1 \times 10^{-6} $
B) $ 1.4 \times 10^7 $
C) $ 3.6 \times 10^{10} $
D) $ 4 \times 10^{-3} $

We are to write them in usual form, i.e., standard decimal form.

- A) $ 4.1 \times 10^{-6} = 0.0000041 $
- B) $ 1.4 \times 10^7 = 14,000,000 $
- C) $ 3.6 \times 10^{10} = 36,000,000,000 $
- D) $ 4 \times 10^{-3} = 0.004 $

But since the question says “Write following numbers in usual form” but doesn't specify which one to pick, and it gives multiple choices, likely it's asking us to identify which number is already in scientific notation or perhaps select one that matches a given value.

But since no specific number is mentioned, maybe it's just asking us to understand how to convert — perhaps this is part of a larger context.

But since it says "write in usual form", and lists options, probably we're supposed to interpret which one is being asked.

Wait — actually, this seems like a multiple-choice question where you choose the correct scientific notation representation of a number in usual form? But the wording is unclear.

Alternatively, maybe it's asking: Which of the following is equal to some number in usual form?

But without a base number, we can’t proceed.

Possibly, this is a formatting issue. Let's skip for now and come back.

But if the task is just to write them in usual form, then:

- A) 0.0000041
- B) 14,000,000
- C) 36,000,000,000
- D) 0.004

So unless more context is given, we can only say: These are their usual forms.

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(3) Find a positive rational number solution of:

$$
\sqrt{21 + 12\sqrt{3}} + \sqrt{21 - 12\sqrt{3}}
$$

Let:
$$
x = \sqrt{21 + 12\sqrt{3}} + \sqrt{21 - 12\sqrt{3}}
$$

Square both sides:

$$
x^2 = \left(\sqrt{21 + 12\sqrt{3}} + \sqrt{21 - 12\sqrt{3}}\right)^2
= (21 + 12\sqrt{3}) + (21 - 12\sqrt{3}) + 2\sqrt{(21 + 12\sqrt{3})(21 - 12\sqrt{3})}
$$

Simplify:

$$
x^2 = 42 + 2\sqrt{(21)^2 - (12\sqrt{3})^2}
= 42 + 2\sqrt{441 - 144 \cdot 3}
= 42 + 2\sqrt{441 - 432}
= 42 + 2\sqrt{9}
= 42 + 2 \cdot 3 = 42 + 6 = 48
$$

So:
$$
x^2 = 48 \Rightarrow x = \sqrt{48} = 4\sqrt{3}
$$

But wait — this is not rational. The problem asks for a positive rational number solution.

But $ x = \sqrt{48} = 4\sqrt{3} $ is irrational.

Wait — did we make a mistake?

Wait — let's double-check.

We have:
$$
x = \sqrt{21 + 12\sqrt{3}} + \sqrt{21 - 12\sqrt{3}}
\Rightarrow x^2 = 42 + 2\sqrt{(21)^2 - (12\sqrt{3})^2}
= 42 + 2\sqrt{441 - 432} = 42 + 2\sqrt{9} = 42 + 6 = 48
\Rightarrow x = \sqrt{48} = 4\sqrt{3}
$$

But $ 4\sqrt{3} $ is not rational.

So why does the question ask for a rational solution?

Wait — perhaps the expression simplifies to a rational number?

But our result is irrational.

Wait — could the expression be equal to a rational number?

Let’s suppose:
$$
\sqrt{21 + 12\sqrt{3}} = \sqrt{a} + \sqrt{b}
$$

Try to express $ \sqrt{21 + 12\sqrt{3}} $ as $ \sqrt{m} + \sqrt{n} $

Assume:
$$
\sqrt{21 + 12\sqrt{3}} = \sqrt{a} + \sqrt{b}
\Rightarrow (\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab} = 21 + 12\sqrt{3}
$$

So:
- $ a + b = 21 $
- $ 2\sqrt{ab} = 12\sqrt{3} \Rightarrow \sqrt{ab} = 6\sqrt{3} \Rightarrow ab = 36 \cdot 3 = 108 $

So solve:
- $ a + b = 21 $
- $ ab = 108 $

Quadratic: $ t^2 - 21t + 108 = 0 $

Discriminant: $ 441 - 432 = 9 $

$ t = \frac{21 \pm 3}{2} = 12 $ or $ 9 $

So $ a = 12, b = 9 $

Thus:
$$
\sqrt{21 + 12\sqrt{3}} = \sqrt{12} + \sqrt{9} = 2\sqrt{3} + 3
$$

Similarly,
$$
\sqrt{21 - 12\sqrt{3}} = \sqrt{12} - \sqrt{9} = 2\sqrt{3} - 3 \quad \text{? Wait — but this is negative!}
$$

No — $ \sqrt{21 - 12\sqrt{3}} $ must be positive.

But $ 2\sqrt{3} \approx 3.464 $, so $ 2\sqrt{3} - 3 \approx 0.464 > 0 $, okay.

But let's check:
$$
(2\sqrt{3} - 3)^2 = (2\sqrt{3})^2 - 2 \cdot 2\sqrt{3} \cdot 3 + 9 = 12 - 12\sqrt{3} + 9 = 21 - 12\sqrt{3}
$$

Yes! So:
$$
\sqrt{21 - 12\sqrt{3}} = 2\sqrt{3} - 3 \quad \text{(since it's positive)}
$$

Now add:
$$
\sqrt{21 + 12\sqrt{3}} + \sqrt{21 - 12\sqrt{3}} = (3 + 2\sqrt{3}) + (2\sqrt{3} - 3) = 4\sqrt{3}
$$

Again, $ 4\sqrt{3} $ — still irrational.

But the problem says "find a positive rational number solution".

Wait — maybe I misread.

Is it possible that the expression equals a rational number?

But we got $ 4\sqrt{3} $, which is irrational.

Unless the problem meant something else.

Wait — perhaps the expression is equal to a rational number, but our calculation shows otherwise.

Wait — maybe it's asking for a value of the expression, and we’re to simplify it.

But $ 4\sqrt{3} $ is not rational.

Wait — perhaps the expression is $ \sqrt{21 + 12\sqrt{3}} \times \sqrt{21 - 12\sqrt{3}} $? That would be:

$$
\sqrt{(21 + 12\sqrt{3})(21 - 12\sqrt{3})} = \sqrt{21^2 - (12\sqrt{3})^2} = \sqrt{441 - 432} = \sqrt{9} = 3
$$

Ah! So the product is 3, a rational number.

But the problem says sum.

Let me recheck the original:

> (3) Find a positive rational number solution of $ \sqrt{21+12\sqrt{3}} + \sqrt{21-12\sqrt{3}} $

But we found it to be $ 4\sqrt{3} $, irrational.

So either:
- There's a typo in the problem.
- Or we made a mistake.

Wait — earlier we said:
$$
\sqrt{21 + 12\sqrt{3}} = 3 + 2\sqrt{3}, \quad \sqrt{21 - 12\sqrt{3}} = 2\sqrt{3} - 3
$$

But $ 2\sqrt{3} - 3 \approx 3.464 - 3 = 0.464 $, positive — okay.

Sum: $ 3 + 2\sqrt{3} + 2\sqrt{3} - 3 = 4\sqrt{3} $

Yes.

But $ 4\sqrt{3} $ is not rational.

So the sum is not rational.

But the problem says "find a positive rational number solution" — implying that the expression evaluates to a rational number.

So contradiction.

Unless the problem is asking for something else.

Wait — perhaps it's asking to find a rational number that satisfies some equation involving this expression?

But it says: "Find a positive rational number solution of [this expression]"

That phrasing is odd.

Perhaps it's a typo, and it should be the product, which is 3.

Or maybe it's asking to simplify the expression and report the result.

But the answer is $ 4\sqrt{3} $, not rational.

Wait — another idea: maybe $ \sqrt{21 + 12\sqrt{3}} $ can be written as $ a + b\sqrt{3} $, and similarly for the other.

Let’s try:

Let $ \sqrt{21 + 12\sqrt{3}} = a + b\sqrt{3} $

Square both sides:
$$
a^2 + 2ab\sqrt{3} + 3b^2 = 21 + 12\sqrt{3}
\Rightarrow (a^2 + 3b^2) + 2ab\sqrt{3} = 21 + 12\sqrt{3}
$$

So:
- $ a^2 + 3b^2 = 21 $
- $ 2ab = 12 \Rightarrow ab = 6 $

Try integer values:
- $ a = 3, b = 2 $: $ a^2 + 3b^2 = 9 + 12 = 21 $ ✔

So $ \sqrt{21 + 12\sqrt{3}} = 3 + 2\sqrt{3} $

Similarly, $ \sqrt{21 - 12\sqrt{3}} = 3 - 2\sqrt{3} $? No — because $ 3 - 2\sqrt{3} \approx 3 - 3.464 = -0.464 $, negative.

But square root is positive.

So instead, $ \sqrt{21 - 12\sqrt{3}} = |3 - 2\sqrt{3}| $, but that's $ 2\sqrt{3} - 3 $? No, $ 2\sqrt{3} - 3 \approx 0.464 $, positive.

But $ (2\sqrt{3} - 3)^2 = 12 - 12\sqrt{3} + 9 = 21 - 12\sqrt{3} $ — yes.

So $ \sqrt{21 - 12\sqrt{3}} = 2\sqrt{3} - 3 $

Then sum:
$$
(3 + 2\sqrt{3}) + (2\sqrt{3} - 3) = 4\sqrt{3}
$$

Same result.

So the sum is $ 4\sqrt{3} $, irrational.

But the problem says "find a positive rational number solution".

Maybe it's a trick — or typo.

Perhaps the expression is $ \sqrt{21 + 12\sqrt{3}} \times \sqrt{21 - 12\sqrt{3}} $, which is $ \sqrt{(21)^2 - (12\sqrt{3})^2} = \sqrt{441 - 432} = \sqrt{9} = 3 $

And 3 is rational.

So likely, the problem meant product, not sum.

Or perhaps the expression is different.

Alternatively, maybe the problem is to find a rational number that is a solution to an equation involving this, but it's not specified.

Given the ambiguity, and since the sum is irrational, but the product is 3 (rational), I suspect the intended expression was the product.

So assume the problem meant:

> Find a positive rational number solution of $ \sqrt{21+12\sqrt{3}} \times \sqrt{21-12\sqrt{3}} $

Then:
$$
= \sqrt{(21+12\sqrt{3})(21-12\sqrt{3})} = \sqrt{21^2 - (12\sqrt{3})^2} = \sqrt{441 - 432} = \sqrt{9} = 3
$$

✔ Answer: 3

So likely, the intended answer is 3.

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(4) Simplify $ \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} $

Rationalize the denominator:

Multiply numerator and denominator by $ \sqrt{3} - \sqrt{2} $:

$$
\frac{(\sqrt{3} - \sqrt{2})^2}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})} = \frac{3 - 2\sqrt{6} + 2}{3 - 2} = \frac{5 - 2\sqrt{6}}{1} = 5 - 2\sqrt{6}
$$

Wait — $ (\sqrt{3} - \sqrt{2})^2 = 3 - 2\sqrt{6} + 2 = 5 - 2\sqrt{6} $

Denominator: $ 3 - 2 = 1 $

So:
$$
\frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} + \sqrt{2}} = 5 - 2\sqrt{6}
$$

But let's verify numerically:

- $ \sqrt{3} \approx 1.732 $, $ \sqrt{2} \approx 1.414 $
- Numerator: $ 1.732 - 1.414 = 0.318 $
- Denominator: $ 1.732 + 1.414 = 3.146 $
- Ratio: $ 0.318 / 3.146 \approx 0.101 $

Now $ 5 - 2\sqrt{6} \approx 5 - 2*2.449 = 5 - 4.898 = 0.102 $ — close.

So correct.

✔ Answer: $ 5 - 2\sqrt{6} $

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(5) What is the unit’s digit in $ (237)^{109} \times (237)^{36} $?

First, combine exponents:
$$
(237)^{109} \times (237)^{36} = (237)^{145}
$$

We need the unit’s digit of $ 237^{145} $, which depends only on the unit’s digit of 237, which is 7.

So we need the unit’s digit of $ 7^{145} $

Pattern of powers of 7:
- $ 7^1 = 7 $ → 7
- $ 7^2 = 49 $ → 9
- $ 7^3 = 343 $ → 3
- $ 7^4 = 2401 $ → 1
- $ 7^5 = 16807 $ → 7 → cycle repeats every 4

Cycle: 7, 9, 3, 1 (length 4)

Find $ 145 \mod 4 $:

$ 145 \div 4 = 36 \times 4 = 144 $, remainder 1

So $ 7^{145} $ has same unit digit as $ 7^1 $ → 7

✔ Answer: 7

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(6) $ \left(\frac{1}{2}\right)^2 \times \left(\frac{-2}{3}\right)^2 = ? $

Compute each:

- $ \left(\frac{1}{2}\right)^2 = \frac{1}{4} $
- $ \left(\frac{-2}{3}\right)^2 = \frac{4}{9} $

Multiply:
$$
\frac{1}{4} \times \frac{4}{9} = \frac{4}{36} = \frac{1}{9}
$$

✔ Answer: $ \frac{1}{9} $

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(7) Simplify $ (x^2 + y^2)^{-1} $

This is $ \frac{1}{x^2 + y^2} $

Now look at options:

a) $ \frac{x^2y^2}{x^2 + y^2} $ ✘
b) $ \frac{x^2 + y^2}{x^2y^2} $ ✘
c) $ \frac{xy}{x^2 + y^2} $ ✘
d) $ \frac{x^2 + y^2}{xy} $ ✘

None of them is $ \frac{1}{x^2 + y^2} $

Wait — perhaps the expression is $ (x^2 + y^2)^{-1} $, which is $ \frac{1}{x^2 + y^2} $

But none of the options match.

Unless there's a typo.

Wait — perhaps it's $ (x + y)^{-2} $? But no.

Alternatively, maybe it's $ (x^2 + y^2)^{-1} $, and they want it as a fraction.

But none of the options is $ \frac{1}{x^2 + y^2} $

Let’s check again:

a) $ \frac{x^2y^2}{x^2 + y^2} $ — no
b) $ \frac{x^2 + y^2}{x^2y^2} $ — reciprocal of a
c) $ \frac{xy}{x^2 + y^2} $ — no
d) $ \frac{x^2 + y^2}{xy} $ — no

So none match $ \frac{1}{x^2 + y^2} $

But perhaps the expression is $ (x^2 + y^2)^{-1} = \frac{1}{x^2 + y^2} $, and that’s the simplified form.

But since no option matches, maybe it's a typo.

Alternatively, perhaps the expression is $ (x^2 + y^2)^{-1} $, and they want to see if it's equal to any of these — but none are.

Wait — maybe it's $ (x^2 + y^2)^{-1} $, and they expect us to recognize it as $ \frac{1}{x^2 + y^2} $, and if that's not an option, then none.

But let’s suppose the expression is $ (x^2 + y^2)^{-1} $, and the correct answer is none.

But since it's multiple choice, and none match, perhaps it's a mistake.

Alternatively, maybe the expression is $ (x + y)^{-2} $, but no.

Wait — perhaps the question is to simplify $ (x^2 + y^2)^{-1} $, and the answer is simply $ \frac{1}{x^2 + y^2} $, but since it's not listed, we can't choose.

But looking at options, none are correct.

Wait — perhaps the expression is $ (x^2 + y^2)^{-1} $, and they want to write it as a fraction, but all options are wrong.

So likely, the intended expression might be different.

Alternatively, maybe it's $ (x^2 + y^2)^{-1} $, and the answer is none, but since it's not an option, perhaps we skip.

But let's move on.

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(8) If $ a^x = \sqrt{b}, b^y = \sqrt[3]{c}, c^z = \sqrt{a} $, find the value of $ xyz $

We have:
1. $ a^x = b^{1/2} $
2. $ b^y = c^{1/3} $
3. $ c^z = a^{1/2} $

Take logarithms or substitute.

From (1): $ a^x = b^{1/2} \Rightarrow b = a^{2x} $

From (2): $ b^y = c^{1/3} \Rightarrow c = b^{3y} = (a^{2x})^{3y} = a^{6xy} $

From (3): $ c^z = a^{1/2} \Rightarrow (a^{6xy})^z = a^{1/2} \Rightarrow a^{6xyz} = a^{1/2} $

So:
$$
6xyz = \frac{1}{2} \Rightarrow xyz = \frac{1}{12}
$$

✔ Answer: $ \frac{1}{12} $ → Option c

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(9) For each positive integer n, which of the following is also a square number?

a) $ n^2 + 2n + 2 $
b) $ n^2 + 8n + 4 $
c) $ n^2 + 2n + 1 $
d) $ n^2 + 6n $

Check each:

a) $ n^2 + 2n + 2 = (n+1)^2 + 1 $ — not a perfect square (e.g., n=1: 1+2+2=5, not square)

b) $ n^2 + 8n + 4 = (n+4)^2 - 12 $ — not always square

c) $ n^2 + 2n + 1 = (n+1)^2 $ — always a perfect square

d) $ n^2 + 6n = n(n+6) $ — not always square

✔ Answer: c

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Final Answers Summary:

1. $ 3.8 \times 10^{11} + 3 \times 10^{12} = 3.38 \times 10^{12} $
2. Usual forms: A) 0.0000041, B) 14,000,000, C) 36,000,000,000, D) 0.004
3. Likely typo; if product, answer is 3
4. $ 5 - 2\sqrt{6} $
5. Unit’s digit: 7
6. $ \frac{1}{9} $
7. None of the options match $ \frac{1}{x^2 + y^2} $ — possibly error
8. $ xyz = \frac{1}{12} $ → Option c
9. Option c: $ n^2 + 2n + 1 $

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✔ Final Boxed Answers:

(3) Likely 3 (if product)
(4) $ \boxed{5 - 2\sqrt{6}} $
(5) $ \boxed{7} $
(6) $ \boxed{\frac{1}{9}} $
(8) $ \boxed{\frac{1}{12}} $ → c
(9) $ \boxed{c} $

Let me know if you'd like clarification on any part.