Question 7

We will examine each statement about powers of numbers and determine if it is always true, only sometimes true, or never true.

Step 1 — Cube numbers and square numbers

Let us consider what a cube number and a square number are. A cube number is a number we get by multiplying an integer by itself three times, like $n^3$. A square number is a number we get by multiplying an integer by itself two times, like $m^2$.

For a number to be both a cube and a square, its exponent in its prime factorization must be a multiple of both 3 and 2. The smallest common multiple of 3 and 2 is 6. So, numbers like $k^6$ are both cube and square numbers.

Let us test some cube numbers.

$1^3 = 1$

Is 1 a square number? Yes, $1^2 = 1$.

$2^3 = 8$

Is 8 a square number? No, because $2^2 = 4$ and $3^2 = 9$.

$4^3 = 64$

Is 64 a square number? Yes, $8^2 = 64$.

Since some cube numbers (like 8) are not square numbers, but others (like 1 and 64) are, the statement is not always true. It is also not never true.

$\boxed{\text{Only Sometimes True}}$

Step 2 — Fourth powers and square numbers

Let us consider a number that is a fourth power. This means it can be written as $n^4$ for some integer $n$. We want to see if this number is always a square number.

We can use the rules of exponents. The rule $(a^b)^c = a^{b \times c}$ helps us here.

We can rewrite $n^4$ like this:

$n^4 = (n^2)^2$

Here, $n^2$ is an integer if $n$ is an integer. So, $(n^2)^2$ means we are squaring the integer $n^2$. This makes $n^4$ a perfect square.

For example, $2^4 = 16$. We know $16 = 4^2$. Here, $n=2$, so $n^2 = 2^2 = 4$. So $2^4 = (2^2)^2 = 4^2$.

This works for any integer $n$.

$\boxed{\text{Always True}}$

Step 3 — Divisibility of fifth power by cube

Let us take any number and call it $a$. Its fifth power is $a^5$. Its cube is $a^3$. We want to know if $a^5$ is always divisible by $a^3$.

Divisible means that when we divide $a^5$ by $a^3$, we get a whole number (an integer). We assume $a$ is not zero, because division by zero is not allowed.

We use the rule of exponents for division: $a^m / a^n = a^{m-n}$.

Let us divide $a^5$ by $a^3$:

$\frac{a^5}{a^3} = a^{5-3}$

$= a^2$

Since $a$ is an integer (and not zero), $a^2$ will always be an integer. This means $a^5$ can be written as $a^3 \times a^2$. So, $a^3$ is a factor of $a^5$.

Therefore, $a^5$ is always divisible by $a^3$.

$\boxed{\text{Always True}}$

Step 4 — Product of two cube numbers

Let us take two cube numbers. We can write them as $x^3$ and $y^3$ for some integers $x$ and $y$. We want to find their product.

The product is $x^3 \times y^3$.

We use another rule of exponents: $(ab)^n = a^n b^n$. This rule also works in reverse: $a^n b^n = (ab)^n$.

So, we can write the product as:

$x^3 \times y^3 = (x \times y)^3$

Let us say $P = x \times y$. Since $x$ and $y$ are integers, their product $P$ is also an integer.

So, the product of the two cube numbers is $P^3$. This is a cube number.

For example, $2^3 = 8$ and $3^3 = 27$. Their product is $8 \times 27 = 216$. We know that $216 = 6^3$. Here, $P = 2 \times 3 = 6$.

This is always true.

$\boxed{\text{Always True}}$

Step 5 — $q^{46}$ as a 4th power and a 6th power

We are given the number $q^{46}$, where $q$ is a prime number. We need to check if it is both a 4th power and a 6th power.

For a number to be a 4th power, its exponent must be a multiple of 4. This means we should be able to divide the exponent by 4 and get a whole number.

Let us check if 46 is a multiple of 4:

$46 \div 4 = 11 \text{ with a remainder of } 2$

Since there is a remainder, 46 is not a multiple of 4. So, $q^{46}$ is not a 4th power.

For a number to be a 6th power, its exponent must be a multiple of 6. This means we should be able to divide the exponent by 6 and get a whole number.

Let us check if 46 is a multiple of 6:

$46 \div 6 = 7 \text{ with a remainder of } 4$

Since there is a remainder, 46 is not a multiple of 6. So, $q^{46}$ is not a 6th power.

Since $q^{46}$ is neither a 4th power nor a 6th power, it cannot be both.

$\boxed{\text{Never True}}$

Answer

(i) Only Sometimes True (ii) Always True (iii) Always True (iv) Always True (v) Never True