Interesting Indices in Incredible Instances! Quiz

Answer: It is squared

If we take the example that I have provided in the question above, 5^2. This means that the quantity 5 has been raised to the power of 2. This is equal to 5 x 5 which equals 25. It has therefore been squared. This really is the basis for all quantities using indices and it is usually the first index or exponent that is taught to children in early mathematical and scientific education.

Answer: 4

If we write this out algebraically, it will look similar to: 4^x = 256. If you didn't have a calculator to hand to work out all different roots (square, cube etc.) then it would possibly be easier to look at this from a different perspective. As you are looking for how many times the integer four has been raised to a certain power you can keep on multiplying 4 by itself until you reach the product of 256. So, 4 x 4 (4^2) = 16; 16 x 4 (4^3) = 64; 64 x 4 (4^4) = 256. Eh voilą, after a few steps you arrive at the product 256 which is equal to 4 raised to the power of 4 (4^4). If you have a better understanding of mathematics, this whole process is far easier with the use of logarithms.

Answer: It is inverted

In mathematics, inversion is very important and can be used in many different disciplines, including calculus. Inversion is sometimes described as taking the reciprocal. Basically, inversion is the process of flipping a function upside down. There is one main example which illustrates this point. Firstly, if we take the integer 4 and raise it to the power -3, you are basically multiplying 4 to the inverse of 3.

This is written in standard notation as 4^-3. This example is sometimes used in British mathematical examinations to trip students up or to allow the students to show that they have a decent grasp of arithmetical methods. If you take the integer 4, how could you flip this upside down so that four ends up as the denominator of a fraction? Well, as it stands you can't as you need to introduce a denominator so that four can become a numerator. With prior knowledge you should, hopefully, come to the conclusion that 4 = 4/1.

As things stand we have attained the equation (4/1)^-3. So, we will now invert 4/1 which therefore becomes 1/4. By doing this you have eradicated the negative sign of the exponent. We are left with 1/(4^3). You now cube the integer four as the exponent suggests: 4 x 4 x 4 = 64. Thus, the answer to the problem 4^-3 is 1/64.

It seems long winded in writing, however, in practice it is much quicker.

Answer: Square root

In mathematics, indices are frequently seen in fractional form. Basic mathematics teaches us that the bottom of a fraction is called the denominator whilst the top is called the numerator. If there is a fractional exponent, in this case 3/2, then a two step process is required to find the solution. You can work out the solutions by using either the denominator or numerator first. Generally, there is one part of the fraction that is easier to use than the other and with an acquired mathematical intuition you can generally spot the easiest method straight off. If we follow the given example, 25^3/2, three is the numerator.

The numerator tells us how many times you need to multiply the number (25) by itself. In this case it would be 25^3 = 25 x 25 x 25.

This would be a difficult way to begin a relatively easy question. The denominator in the same example is two. As the question has shown us, the denominator shows us how many times you need to root a number. Hence, 25^2 = square root (because it is being rooted twice) of 25.

The square root of 25 is significantly easier to work out than 25 x 25 x 25. Ergo, intuitively you would use the denominator function first. The square root of 25 is 5 and you are subsequently left with 5^3. Using basic index notation, 5^3 = 5 x 5 x 5 = 125.

Answer: 1/3

The denominator of the exponent of a base always stands for the root function. These are as follows; base^1/2 = square root, base^1/3 = cubed root, base^1/4 = fourth root and so on and so forth. The numerator always stands for the raising power function. So, base^2/3 = (cubed root of base)^squared.

The presence of the negative in the exponent does not change this, except that the base is inverted.

Answer: Add them (4 + 2)

At first you would be justified in thinking that you would multiply them, but in fact you do add them, strange? Well if we consider this question algebraically, a^4 x a^2 = a^4+2 = a^6. Now if we assign a randomly generated number for a, you will find that the addition of indices makes sense. If we let a = 5: 5^4 x 5^2 = 625 x 25 = 15625. To prove that the addition of exponents is mathematically correct we can find out what 5^6 is equal too. If it matches we know it is right. So if we take 5^6 we discover that it does indeed equal 15625! So, the rule is, when you are multiplying indices, you add the exponents.

Answer: [25(a^6)]

When you are presented with an alpha-numeric base, you need to deal with the number and the algebraic letter individually in order to attain the correct answer. The first thing that needs to be done is to recognise the exponent which in this case is just base^2 (base squared). If you deal with the number first you find that 5^2 = 25.

This rules out two of the above options already as we know that the coefficient of the a^unknown is 25. The next step is to work out the algebraic portion of the question: (a^3)^2.

This is an instance in which you multiply the two indices. This is because (a^3)^2 is equal to a^3 x a^3. When you multiply indices you are actually adding them together, so, a^3 x a^3 = a^3+3 = a^6. The final part of the question is to put the two pieces back together to obtain the answer [25(a^6)].

Answer: ([a^4][b^6][c^3])

What I generally find is that people struggle using the square root function with algebraic indices. The way to look at this is from a beginner's point of view is to revert back to other rules of index notation. If you have (a^4)^2, which is using the squaring function, you multiply the two indices which results in a^8. Most people generally know how to get this with relative ease.

However, when it comes to using the square root function, a lot of people look at it with a blank expression! Despite this, it is quite simply using the same principle as that of which you have already used.

The square root of a^8 is a case of again multiplying the two indices. Square root can be written as ^1/2. So, (a^8)^1/2 = 8 x 1/2 = 4. Therefore, the square root of a^8 = a^4. If you look at the answer to the above question, you will see that this pattern is followed throughout the expression.

Answer: base^1

Mathematics is always about trying to find the easiest routes to attain answers to expressions and equations. If things weren't simplified the mathematics would be difficult to read and mistakes will arise. It is for this reason that such things as 10/1 are shown as just the integer 10 and, using the example above, 5^1 is just shown as the integer 5. Mathematics is all about accuracy and clarity.

Answer: 1

One of the few rules that has to be learned is that all bases raised to the power of 0 will equal 1. If we take this situation where a is the base and it is equal to 5: (5^3/5^3). In this example you are dividing 5^3 by itself. Using laws of indices you subtract the exponents during division, so, 5^3/5^3 = 5^(3-3) = 5^0 = 1. If you look at the example again, you will see that when you divide a number exactly by itself you will end up with 1 as your result. To prove this further, if you expand the expressions, 5^3/5^3 = 125/125 = 1.

Well I hope you enjoyed this quiz; thanks for taking the time to play it!