4^3x7^-2)^-5/ 7^9
To simplify the expression 4^3 * 7^-2)^-5 / 7^9, we can start by simplifying the inside parentheses first.
Inside the parentheses, we have 4^3 * 7^-2.
First, evaluate 4^3:
4^3 = 4 * 4 * 4 = 64.
Next, evaluate 7^-2:
7^-2 = 1 / 7^2 = 1 / 49.
Now, the expression inside the parentheses simplifies to 64 * (1/49).
Let's write the expression again: (64 * 1/49)^-5 / 7^9.
Now, simplify (64 * 1/49)^-5:
(64 * 1/49)^-5 = (64 / 49)^-5.
Since the base is the same, we can simplify by multiplying the exponents:
(64 / 49)^-5 = 64^-5 / 49^-5.
Evaluate 64^-5:
64^-5 = 1 / 64^5 = 1 / (64 * 64 * 64 * 64 * 64) = 1 / 1073741824.
Evaluate 49^-5:
49^-5 = 1 / 49^5 = 1 / (49 * 49 * 49 * 49 * 49) = 1 / 5764801.
Now, the expression simplifies to (1 / 1073741824) / (1 / 5764801).
To divide by a fraction, we can multiply by the reciprocal of the fraction:
(1 / 1073741824) / (1 / 5764801) = (1 / 1073741824) * (5764801 / 1).
Multiplying the fractions gives us:
(1 * 5764801) / (1073741824 * 1) = 5764801 / 1073741824.
Finally, simplify 7^9:
7^9 = 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 * 7 = 40353607.
Therefore, the simplified expression is 5764801 / 1073741824 / 40353607.
To simplify the expression (4^3 x 7^-2)^-5 / 7^9, we can split it into two parts: the numerator and the denominator.
First, let's simplify the numerator: (4^3 x 7^-2)^-5.
Step 1: Evaluate the exponent for 4 and 7.
4^3 = 4 x 4 x 4 = 64
7^-2 = 1 / 7^2 = 1 / (7 x 7) = 1 / 49
Step 2: Rewrite the expression with the simplified values:
(64 x 1/49)^-5
Step 3: Simplify the numerator further:
64 x 1/49 = 64/49
Step 4: Apply the negative exponent:
(64/49)^-5 = 49^5 / 64^5
Now, let's simplify the denominator: 7^9.
Step 5: Evaluate the exponent for 7:
7^9 = 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 = 403,536,07
So the expression (4^3 x 7^-2)^-5 / 7^9 simplifies to:
(49^5 / 64^5) / 403,536,07
To simplify the expression (4^3 x 7^-2)^-5 / 7^9, we can start by simplifying the terms inside the parentheses.
First, let's deal with the exponents.
4^3 = 4 x 4 x 4 = 64 (because raising a number to the power of 3 means multiplying the number by itself three times).
On the other hand, 7^-2 means we have to take the reciprocal of 7^2.
7^2 = 7 x 7 = 49, so taking the reciprocal gives us 1/49.
Now, we can substitute these values back into the expression:
(64 x 1/49)^-5 / 7^9
Next, let's simplify further by applying the exponent of -5 to the terms inside the parentheses (64 x 1/49).
To raise a value to a negative power, we can take the reciprocal.
So, (64 x 1/49)^-5 = (64/49)^-5.
Now, let's simplify 7^9:
7^9 = 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 = 40353607 (because raising a number to the power of 9 means multiplying the number by itself nine times).
Now we can substitute these values back into the expression:
(64/49)^-5 / 40353607.
To simplify further, we can apply the negative exponent (-5) to both the numerator and denominator.
The negative exponent means we take the reciprocal.
So, (64/49)^-5 is equal to 49/64 raised to the power of 5, or (49/64)^5.
Now we have:
(49/64)^5 / 40353607.
To raise a fraction to a power, we can raise the numerator and denominator separately.
(49/64)^5 = 49^5 / 64^5.
This gives us:
49^5 / 64^5 / 40353607.
To calculate 49^5 and 64^5, simply raise each number to the power of 5:
49^5 = 49 x 49 x 49 x 49 x 49 = 5764801.
64^5 = 64 x 64 x 64 x 64 x 64 = 1073741824.
Now we can substitute these values back into the expression:
5764801 / 1073741824 / 40353607.
To divide by a fraction, we can invert the divisor and multiply:
5764801 / 1073741824 x 1 / 40353607.
Simplifying:
5764801 / 43383880184810048.
Therefore, the final simplified expression is 5764801 / 43383880184810048.