Solve: 10^8 + 10^-4 + (1/10)^-16 Expression with Mixed Powers

Exponent Laws with Negative Powers

108+104+(110)16=? 10^8+10^{-4}+(\frac{1}{10})^{-16}=\text{?}

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Simplify the following problem
00:03 In order to remove a negative exponent
00:06 Flip the numerator and denominator and the exponent will become positive
00:10 We'll apply this formula to our exercise
00:27 The same formula applies to fractions as well
00:30 Flip the numerator and the denominator and we should obtain a positive exponent
00:34 We'll apply this formula to our exercise
00:42 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

108+104+(110)16=? 10^8+10^{-4}+(\frac{1}{10})^{-16}=\text{?}

2

Step-by-step solution

Let's use the law of exponents for negative exponents:

an=1an a^{-n} = \frac{1}{a^n} and apply this law to the problem:

108+104+(110)16=108+1104+(101)16 10^8+10^{-4}+(\frac{1}{10})^{-16}=10^8+\frac{1}{10^4}+(10^{-1})^{-16} when we apply the above law of exponents to the second term in the sum, and the same law but in the opposite direction - we'll apply it to the fraction inside the parentheses of the third term in the sum,

Now let's recall the law of exponents for exponent of an exponent:

(am)n=amn (a^m)^n=a^{m\cdot n} we'll apply this law to the expression we got in the last step:

108+1104+(101)16=108+1104+10(1)(16)=108+1104+1016 10^8+\frac{1}{10^4}+(10^{-1})^{-16}=10^8+\frac{1}{10^4}+10^{(-1)\cdot(-16)}=10^8+\frac{1}{10^4}+10^{16} when we apply this law to the third term from the left and then simplify the resulting expression,

Let's summarize the solution steps, we got that:

108+104+(110)16=108+1104+(101)16=108+1104+1016 10^8+10^{-4}+(\frac{1}{10})^{-16}=10^8+\frac{1}{10^4}+(10^{-1})^{-16} =10^8+\frac{1}{10^4}+10^{16} Therefore the correct answer is answer A.

3

Final Answer

108+1104+1016 10^8+\frac{1}{10^4}+10^{16}

Key Points to Remember

Essential concepts to master this topic
  • Negative Exponent Rule: an=1an a^{-n} = \frac{1}{a^n} converts negative powers to fractions
  • Power of Power: (am)n=amn (a^m)^n = a^{m \cdot n} so (101)16=1016 (10^{-1})^{-16} = 10^{16}
  • Check: Verify each term: 108+1104+1016 10^8 + \frac{1}{10^4} + 10^{16} matches our answer ✓

Common Mistakes

Avoid these frequent errors
  • Incorrectly applying negative exponent rule to compound expressions
    Don't treat (110)16 (\frac{1}{10})^{-16} as 11016 \frac{1}{10^{16}} = wrong direction! This ignores that the entire fraction is raised to a negative power. Always first rewrite 110=101 \frac{1}{10} = 10^{-1} , then apply (101)16=1016 (10^{-1})^{-16} = 10^{16} .

Practice Quiz

Test your knowledge with interactive questions

\( \)Choose the corresponding expression:

\( \left(\frac{1}{2}\right)^2= \)

FAQ

Everything you need to know about this question

Why does (110)16 (\frac{1}{10})^{-16} become such a huge number?

+

When you raise a fraction to a negative power, you flip it and make the exponent positive! So (110)16=1016 (\frac{1}{10})^{-16} = 10^{16} - that's 10 multiplied by itself 16 times!

How do I remember when to flip fractions with negative exponents?

+

Remember this pattern: negative exponent = flip and make positive. So (ab)n=(ba)n (\frac{a}{b})^{-n} = (\frac{b}{a})^n . The negative sign tells you to flip the fraction!

Can I add these terms together to get one number?

+

Not easily! 108=100,000,000 10^8 = 100,000,000 and 1016 10^{16} is much larger. The answer stays as 108+1104+1016 10^8 + \frac{1}{10^4} + 10^{16} because these are very different sized numbers.

What's the difference between 104 10^{-4} and (110)16 (\frac{1}{10})^{-16} ?

+

104=1104 10^{-4} = \frac{1}{10^4} is straightforward. But (110)16 (\frac{1}{10})^{-16} needs two steps: first recognize 110=101 \frac{1}{10} = 10^{-1} , then apply (101)16=1016 (10^{-1})^{-16} = 10^{16} .

Why don't we combine the 108 10^8 and 1016 10^{16} terms?

+

You can only combine terms when you're multiplying powers with the same base, not adding them! 108+1016 10^8 + 10^{16} stays as is, while 108×1016=1024 10^8 \times 10^{16} = 10^{24} .

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Exponents Rules questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

More Questions

Click on any question to see the complete solution with step-by-step explanations